Taekwondo does that for you. Force Unarmed as well, but Taekwondo has special moves and blocking.
The issue is that the trick shooting he’s doing requires the bow to have little to no draw strength. It lets you shoot faster, but your range and penetration are worthless in return. In other words, showy and useless in an actual fight.
Hitting things with a bow, the issue is that the bow is fragile. Even if you made a MA that had special martial arts techniques to better hit enemies with bows in melee, the weapon would still get damaged very quickly as the game code stands.
You could make a blade bow or whatever for a mod, but then again, if you do, it’s no different than a bayonet, you can hit with it to your heart’s content without needing an MA.