Your character has a terminal case of “trying too hard”, best to take him out behind the woodshed right now, and put him out of your misery.
Kung Fu is not a martial art, it’s not even a family of martial arts; it’s a collection of unrelated martial arts that originated in China in a specific historical timeframe. Karate is an Okinawan martial art. Using either of these would be an affront to a Japanese hitman or assassin.
A Japanese Assassin would be a Ninja, full stop. They’d practice their family’s variant of Ninjitsu. Practicing Chinese martial arts like Wushu or Shaolin would be a stain on their honor.
A Japanese hitman would, almost certainly be Yakuza. These guys do not mix with Ninjas. To the Ninjas, and for that matter most of Japanese society, the Yakuza are street rats, it would be a disgrace to associate with them. To the Yakuza, the Ninja would be an uncomfortable reminder that their place in modern Japanese society isn’t earned. Also, like Ninjas, Yakuza aren’t going to be learning non-Japanese martial arts, including Karate.
If you’re scratching your head right now and saying, “but, Okinawa is part of Japan”, you’re absolutely right, today. Historically it wasn’t, and the Japanese still look down their noses at its people, their martial forms and weapons.
Here’s the thing; there’s the classic writing advice, “write what you know.” You can think of this as the training wheels of writing, eventually you’ll be researching new things, and writing about stuff you don’t have any background in, but for today, you probably want to trash this whole project and start over with something much smaller and closer to home.
I’d actually say, ditch the violence as well. I mean, from whatever you end up working on. Violence can be a very difficult thing to get right. Start with characters talking to each other, they don’t have to like one another, or agree on anything, but start with dialog. Build your stories in places you understand. It’s not what you want to write, I get that, but it will give you the tools to write what you want to once you’ve learned more about what you’re doing.
Also, writing characters in any culture you’re not intimately familiar with is very difficult. This is especially true of Japan, which, even today, has a very ridged and stratified society, with very strict rules of behavior that change based on context.