You might want to consider it could be put down if it gets to a rescue center but no one adopts it. There are way too many unwanted cats for them all to be homed.
A cat on our street was abandoned by its owner and attempted to move in with me and my flatmate. We did not want a cat. We did not want any pets. We however could not face the risk/guilt of causing it to be killed.
We love the thing to bits now and its been here about 5 years. Its moved a friend in as well in the last 6 months as you might see from a prev post. Its arrival has caused problems with a neighbor and we did not want another cat. Its charm itself and very amusing. It shows no sign of moving out, and strangely now we seem to be glad.
Give up, love the cat – less guilt and a lot of rewarding moments.