I would let it out where the cat had found it if you can't bring yourself to kill it. I hate mice/rats and love the fact that my four cats keep their numbers down but with birds I have found that even when they seem to be dead - leave them in a quiet place for a while and they fly off and carry on with their business.
with my cats - one only catches prey - if i can get to it before damage is done i take them to the local allotments and release them.
if they are shocked but not badly injured ditto.
if they are badly injured i either drown them or leave the cat to finish the job....on occasion ive taken them to the vet to pts,....but if not practicable either finish it quickly or let the cat do it. i dont take them off the cat now if they are injured.....its kinder to let them finish the job and not prolong the agony.
luckily mine done hunt much....i found a shrew live and kicking on the stairs last weekend. my dopey cat was sat watching it without any attempt to hur it....there wasnt a mark on it. he went to the allotments....