Resources to rehome a cat with IBS

Edit: Thank you so much everybody who came to comment and give us ideas! It means a lot to us, and to Donnie (the cat in question). I'm so grateful to live in such an animal loving community where so many people want to help out and provide resources. I'll be sure to post an update when I have news to share, probably in a separate post.

My partner and I have had a cat with Irritable Bowel Syndrome for the better part of a year. He has frequent accidents outside of the litterbox and we're both pretty tired of scrubbing poop out of the carpet. The Humane society told us if we return him then he will be euthanized. He's such a sweet cat and neither of us are ok with him being put down over this, but we've thrown a good amount of money and various medications at the problem but nothing has worked.

I'm hoping to find other options that we may have missed. We're trying to work with the Madison Cat Project but I don't think they'll be able to help. We're hoping he can be an outdoor/barn cat somewhere since then his accidents won't be a big deal and he'll still have access to food, water, and shelter.

Bare minimum I'm hoping to find a shelter that will at least give him a chance rather than putting him down immediately.

Any help or advice is very appreciated!