Such a nightmare decision for you!
Things are so very different over here, so of course I shouldn't really be butting in....
Sorry.
I was a volunteer dog walker for Battersea Dogs + Cats Home (about the biggest in our Country, tv shows etc. about it) until my back packed up and I couldn't do all the miles that were needed anymore -
www.battersea.org.uk - no animal is put to sleep unless there are major medical or behavioural issues, and that's how it generally is here, throughout all of our shelters.
Nobody chooses which animals are or aren't allowed in - they all are.
I really, truly feel for you, and I don't know what I'd do. Good luck in your decision, Sally x