If you've already had the discussion with the player about "Hey, let's see what we can do to give you the feel you're going for without breaking the game", then here are some suggestions:

1) Discuss with the group about changing the way firearms work for your game. If the player wants a 'firearms focused' character for the coolness factor, then changing Firearms to not deal an automatic Dramatic wound or maybe anyone damaged by Firearms take only 1 Automatic Dramatic Wound / combat or something shouldn't have a big impact on the character. If this is met with a lot of whining, then the player is just looking to game the system

2) Discuss with the group that they may encounter enemies with several firearms as well and if everyone is good with the more 'lethal' tone to the game, try it out a few times. Give a brute squad firearms, give a couple of villains multiple firearms. 

3) Make it cost additional Raises (1 / Firearm) to draw, fire and secure each firearm during a Round. This represents the difficulty in wearing so many firearms and using them effectively.

John

Hopefully with these rules my player can still feel useful with some sturdy pistols, while also ensuring that every combat where he has them doesn't become trivial. Probably still a good idea to recommend that he bring a couple of throwing knives or something just in case.

I think this statement is the key...why is the player choosing this fighting style? Is it because it looks cool and feels cinematic to him? If so, then any small change to keep firearms fair with other weapons shouldn't cause him any heartache. He will be just as effective as any other non-duelist character using melee weapons. 

If he's using them BECAUSE they deal Dramatic Wounds and its important that he's killing stuff so easily, that's a more fundamental issue.

John

Another fix for these situations is tweaking villain health.  I would keep the total number of wounds the same, but give the villain a dramatic wound for every four wounds, instead of just four total, with more health between.

Plus one at the end for numbers not divisible by four.

What about social consequences?

A similar problem has cropped up in a number of FFG Star Wars games around PCs running around in combat-grade armor and toting military-grade rifles that feature the very problematic autofire quality; if people think Duelists are worrisome, they've got nothing on a Star Wars PC that's spec'd to abuse the hell out of the Autofire rules.

Running around with several pistols strapped to your person is going to draw attention, and generally NOT the good kind.  Blackbeard "got away with it" because he did it during the times that his ship was actively raiding another vessel.  I doubt he walked around in his spare time with pistols stowed upon his person, especially during those times when his ship was in a safe harbor.

If this PC tries strolling around a civilized area with a bunch of pistols strapped to his body, then have the locals shy away from that person, and the local authorities take a vested interest in what such a heavily-armed person is doing, not unlike how most people and authorites in today's society would react to someone walking around in an armored tac-vest with an assault rifle over their should and plenty of ammo strapped to them; being a Player Character doesn't mean that the character is automatically immune to the law or the consequences of breaking them.  Carrying around a sword and maybe a pistol is more acceptable, since the former are an accepted means of self-defense that requires some degree of training to use properly, while the latter are generally uncommon and take a significant amount of time to reload.

Also, take into account environmental factors.  The firearms of the time are generally flintlocks, and not always the most reliable of weapons.  Don't abuse that, but don't ignore it either; if the PC's recently gotten soaked, there's a decent chance their pistols have also gotten wet and the flint won't spark properly.  I'd suggest spending a Danger Point to invoke that bit of GM Fiat, but again to do it sparingly.

Granted, there's always the "have the bad guys start doing the same thing."  Now as I understand it, a Brute Squad simply does their Strength Rating in Wounds no matter what weapons they are using, but perhaps you could have a Brute Squad that's armed with muskets or pistols, who thus inflict a Dramatic Wound upon the target.  Or have a Villain that uses a similar tactic as the PC, spending their Raises to unload with a bunch of pistols (such El Duce from the first Boondock Saints movie), who may focus several of their shots on the pistol-packing PC due to seeing them as a serious threat due to employing the same tactics (carry a bunch of pistols and fill your target full of lead) and knowing how deadly that can be.  Again, not things that you should do constantly, or even all that frequently, as the last thing you want is to encourage an arms race with this PC to see who can abuse the rules regarding firearms the worst.

If you do opt to make a mechanical change, I would suggest that instead of the automatic Dramatic Wound, that firearms instead deal 3 Wounds.  This will wind up making firearms far less dangerous to major Villains (who can have a Strength that's two or three times that amount) but still a threat to the PCs.

I have a player who wanted to go firearms heavy. I just ignored the bonus damage for firearms - they are just another weapon. But he has to use a raise every round to "not be down to his last shot" - less dickish than out of ammo  altogether. He also has to work extra hard not to commit murder and therefore gain corruption points.

@Jacob, according to the core rules on page 296:

In the 7th Sea game system, characters aren’t killed, they become Helpless. Even gunfire cannot kill a character without a deliberate act from a Hero or a Villain.

Therefore, you Heroes can confidently shoot an enemy directly in the head with multiple and all is perfectly safe, since even in real life not every headshot kills. If the player does not use "murder" as the approach, headshots are perfectly safe, even beheadings, there is that 1% chance the target survives so not murder. 

I might also say that if the pistol attack fails to deal any Wounds, then no Dramatic Wound gets inflicted.  Granted, that's my intrepretation of how it's meant to work; spend your raise(s) with the pistol to inflict Wounds, target chooses how many raises they want to spend to negate the Wounds, and if target took a single Wound, then they also take a Dramatic Wound.

Having thought about it, I don't think a character having a number of pistols is actually all that much of a problem. Blackbeard is commonly depicted wearing something like 6-10 pistols (a brace at each hip and three braces on a bandolier)

The way I'd run it is as follows:

Assuming that the character has the first brace of pistols readied at the start of the round, they must spend 1 raise for each of the two pistols to fire it and hit what they're aiming at, they then need to spend another raise to discard the used pistols, and draw the next two. After that it's rince and repeat. They're unlikely to get through more than 4 shots in a round in absolutely ideal circumstances where the villain and their allies are standing around doing nothing.

I'd say that once the first two raises are spent to shoot at the villain (assuming that no Brutes intercept the shot) the discard and redraw would take place on the characters next action in the running order, e.g. if they had 5 raises, they would spend 2 to make the first shots, then anyone else on 5 and 4 raises would get to go, then they could spend the raise to discard and redraw, and then the further raises to shoot - you could also argue that the 2nd set of shots has to happen on the character's next action after the discard and redraw. That would mean Shoot on 5 raises, discard and redraw on 3 raises, fire 2nd volley on 2 raises. Next round they then have to spend their first raise to discard and redraw their next brace of pistols.

That gives the villain chance to simply get into cover or flee the scene, removing the opportunity for further shots. It also means that - if they are committed to just shooting at the villain and nothing else, the character will have very few (or even 0) raises to avoid any incoming attacks, or avoid any consequences from the situation (such as hitting the hostage, or other PC that the villain is hiding behind), or the environment (pitching deck, panicked horses, slippery floor etc).

You're gonna charge people a Raise to discard a pistol? That's harsh! I mean, is there anything else you'd charge a Raise to drop? A sword? A lantern? A match? Ok, maybe a match but only if it was to light something on FIRE! What about an empty bottle? Or a letter? A glove? 

Now a raise to draw a weapon or pistol from a holster it sheath, sure. But just to discard something shouldn't cost a raise unless there is something more particular to it ("I hand my discard pistol to my trusted lackey and...").

But yeah, if shooting a pistol at an enemy takes three raises to pull off (draw, fire, discard), no one will be using guns in your game.

You might consider doing something similar with Lunges, since they have roughly the same damage output.

Hmmm, your strategic comma suggests otherwise, but ok. Ignore my color commentary. :)

No, no, no! Don't try to make using firearms harder or make them weaker. That will only make players want to use them more. Forbidding something is the fastest way to make players do that ;). 

You WANT your players to defeat villains easily using firearms. I'm serious. Example? Make a scene where they have to choose:

I can spend a Rise and fire to hurt the villain.
        OR
I can spend a Rise to rescue a tied princess that villan have thrown into a lake. And she's drowning.
        OR
I can spend a Rise to dodge the arrows that are being shot at me.

Make so many consequences, risks and opportunities so that they will run out of their raises, and they will HAVE to choose what do they want most. Do they want to hurt the villain so bad? Do they value life of a princess more? "The villain is there, you're so close... You can catch him, ..Only if you  sacrifice the princess..."

Don't make using firearms hard. Make situations where they DON'T want to use firearms.

And then - use Danger Points to increase the needed number to make a set to 15. Activate Villain's abilities and magic. Then IT'S ON. They can have as many pistols as they want, but that won't save the princess. 

Please let me know if that helps you!

You can also use Pressure from the Villain to make it more difficult to target them than the Brutes by having the Villain duck and move behind them.  Melee-oriented characters would still be able to get to them normally, with the Villain focused on the pistolier.

If they recognize the pistolier as the most dangrous adversary, they can order their Brutes to focus on them, having some of the Brutes bypass the melee Heroes if able, or using ranged weapons, even thrown knives and the like.

The Villain can create a Consequence by engaging into melee with a Hero and trying to use them as cover from the Pistolier, or pulling a captive before them.

I'm still a noob so I apologize if I'm out to lunch here.  However, reading the comments here I thought of that Iconic Indiana Jones scene.  The Duelist draws his sword is does some fancy moves with it.  The guy with a gun shoots him. End of Story.  While this is early in the history of fire-arms, fire-arms became so dominant for just the reason that they were so deadly with so little skill required.

 I've also noted that many people have also complained about how deadly duelists are compared to non-duelists. This pistol 'exploit' may be one way to deal with this.

When I start my campaign, I'm going to make clear that we may have to house-rule combat as we go on.

If there is a serious problem with firearms being over-powered, I would use the # of raises option or a fire every other round option (to represent that while the PC has one pistol already primed, the other pistols aren't necessarily ready to go)