Instead of taking Improved Unarmed Strike, you could instead buy Armor Spikes. These do 1d6 damage (for a medium character) and can be used by anyone with proficiency in all Martial Weapons. Alternately, a spiked gauntlet does 1d4 damage (for a medium character) and usable by anyone with Simple Weapon proficiency. You'd probably only put one on the hand that holds the bow, rather than the hand that draws back the string, though.