How to Make a Bow-Kata Monk/Fighter?

[Arkhios]

As to the whole melee bow being an Improvised Weapon thing... Tavern Brawler makes you proficient with Improvised Weapons so you can totally make a Bow (improvised club) or Arrow (Improvised dagger) a Kensei Weapon.

Improvised Weapon is not a Weapon. Period.
Tavern Brawler doesn't change the fact. Meaning, you totally cannot choose an Improvised Weapon as your Kensei Weapon (so, no, you can't choose an Arrow to be a Kensei Weapon).
At least not by RAW. However, I won't argue it wouldn't be cool, and a particularly lenient DM might let you do so anyway, but that would be a house rule.