Do you have Humane Society in your community or feral cat people you can call? I had a stray cat I fed and then had spayed through the Humane Society but then finding a home and all that got complicated too. The feral cats folks were the ones to come to my rescue. If you can get some hands on help maybe you can reason with your neighbor to give you just a little bit more time.
I understand your angst!

ps cats are pretty smart about finding new feeding places but get some help. There should be some out there.