I guess if you're expecting a stock 10/22, let alone a "beat up 10/22", to shoot alongside your Cooper, you might as well flip it & move on.
A lightweight barrel by Kidd, Volquartsen, Tactical Solutions and possibly others, will help as will a good trigger and stock. Or you could get a complete Volquartsen, Kid or Tactical Solutions rifle already tricked out, but you'll be spending multiples of what you have into yours.
BTW, your action is worth more than $50, with alot of people doing custom builds these days it's hard to find a complete action for much less than you paid for your rifle.