If no one wants the cat a no-kill shelter would be the best choice IMO.

As callous as it sounds, failing that I would put him down myself. Some vigorous play to relax him, a hearty meal, a bit of contented lap sitting for a the final minutes and then a quick twist and a snap of the neck. Followed by a dignified burial and, likely, a stiff drink. It is a tough reality. But it is real.

If a friend or relative was facing such poor options with my final disposition I would like to think they would do the same for me. A quick and painless out is about as good as I could expect.

Taking care of such problems in-house at the hands of a loved one is, I think, the responsible way to go and not so bad. A lot of people think you can just let them go and somehow it will magically 'all work out'. Like a Disney afternoon movie. Mostly it doesn't and they are condemning their pet to long term starvation, misery, loneliness, vicious fights with wild or feral animals for food, and a messy and painful death by getting hit by a car, infection and/or a slow death by disease.

I wouldn't want my pet to suffer like that and I don't like the idea of sticking others with my problems or my companion having to face its final minutes frightened, alone and his final exit handled by a stranger. I wouldn't want my pet to suffer because I lacked the intestinal fortitude to do what has to be done and to do it well.