The way I see it, INA and Strong Jaw are the same type of bonus therefore not able to stack with each other. So both of you are wrong as they shouldn't be able to be used at the same time.
Depending on your DM, you should be allowed to use a bow or crossbow as an improvised melee weapon. And being a melee weapon, you can make AoO with it, however it's up to the DM of the game to make the final call on it.
