If it were me, I'd want to have "animal rescue" or "animal adoption" in the name, reason being that when someone does an internet search, you want your name to fit what they will type in as a search. Then I'd start it with a positive adjective word or phrase, the way retirement homes to to keep them from sounding like a prison for old people. Something like "helpful hands animal rescue and adoption".

I think it's admirable what you're seeking to do; I see a lot of these places fail, though. There were two exotic animal rescue shelters in my city; they both shut down because they became overwhelmed with time commitment and financial issues: it's easier to find animals no longer wanted than people who will want them. Best of luck!