As a DM, I've been waiting patiently for a player to ask for improvised weapons as a Kensei weapon, effectively playing Jackie Chan.

I'd do Kensei Monk with 3 levels of Battlemaster for the most simple build. Your "unarmed" strikes are made with the bow (or you can use the bow as an agile parry), or you can "aim" to add another d4 of damage to your ranged hits

However, another thing you could consider is Battlemaster 5 / Rogue X with either Monk 1 or the Tavern Brawler Feat. This gives you something functionally similar as above, but without having to worry about Ki. Instead of having a scaling Martial Arts Die, you'd instead have scaling sneak attack shenanigans. I'd probably go either Swashbuckler or Inquisitive, although a scout could work as well.