Cats brought mouse in

This is the second time my cats have brought a mouse in. The first time it was dead, this time we saw something crawling across our floor. It’s a tiny mouse. He has a broken back leg, (probably from our cats) if I take him outside either my cats will kill it or something else will. My kids think it’s cute and so do it. Would it be wrong if I bought a cage and kept it so it can have a safe place with food and water since it is wild? (We wouldn’t be touching it) Or should I let it outside and let nature course 😩