None. Given the time frame you’re talking about, a character like that would be lucky to have some informal training on how to throw a punch without dislocating their thumb.
About the only thing he could have access to would be bare knuckled boxing. But, it’s still unlikely.
Some of the traditional Chinese martial arts were practiced, but there is literally no way your character could have had access to them. More than that, it would be incredibly unlikely that they’d see any value in them, or seek them out. Eastern styles were viewed as ineffective, poncy, effeminate, or just not real fighting. This is a view that actually persists in some parts of the Western US today. In that environment, your character isn’t likely to see any merit in those styles, much less try to train in them, and even if he wanted to, he’d be still be prevented from doing so.
Most Asian martial arts maintained a very strict prohibition against training students from other ethnicities (including other Asians). This didn’t change until the 1940s and 50s, so well past the timeframe of a conventional western. This also means it would be completely anachronistic to give a character Japanese and Chinese styles.
If your character is Chinese (or another Asian ethnicity), it’s possible they’d have been trained in a traditional style, but they’d also face severe racism, to the point that they would have a hard time finding work as a mercenary.
Okay, one more thing, that I’ve skipped so far because I might be reading too much into this, the whole “distinctive fighting style” phrase sets off all kinds of warning bells for me.
If you’re trying to make your character more unique, stop. You don’t need to make your character special, for them to be compelling. You distinguish a character in your writing, not by stapling “cool” things to them. Your character needs to belong in the world you’ve created, creating one that doesn’t is a recipe for disaster.
Your character is a hired gun, like hundreds or thousands of others. You make him unique by what he says, thinks, and does, not by giving him neat toys or an easy way out of trouble.
So, some recomendations:
If you’ve never read The Dark Tower series by Stephen King, You need to. It’s more of a horror Western (actually, most of these suggestions will be, but… still.)
Deadlands was an alternate history RPG set in a world where the Civil War rages on, and supernatural forces have been unleashed on the world. As with The Dark Tower this is more horror than straight up fantasy.
Ravenous is a vampire movie that’s not, quite, about vampires. Set in the early 19th century Sierra Nevadas, it focuses on a single, isolated, fort. Again, more horror than fantasy, but it should give you some ideas to play with.
Yojimbo is, functionally, a western set in Edo period Japan. As with a lot of Kurosawa’s work, it’s been adapted repeatedly. If you don’t see this as useful, you might want to look at Last Man Standing. It’s not nearly as good, but it might give you some ideas.
Arcanum: Of Steamworks and Magick Obscura isn’t a western… technically. There are a few minor western themes in a few locations, but that’s not the point. This was a massive RPG from Tim Cain, one of the original designers of the Fallout franchise. The setting takes a Tolkien style high fantasy world and turns the clock forward to 1885. Part steampunk, part political commentary, this game will give you a lot of stuff to chew on. If you’re talking about a high fantasy world hitting the industrial revolution, this is a must play.