The vast majority of countries do not recognise emotional support animals, let alone give them special status. You'll even struggle getting some places to recognise a guide dog or hearing dog. Quarantine absolutely applies to pets, especially for things like avian flu which only affects birds. If you visit a country where there is a highly pathogenic strain of avian flu, Oliver is definitely going to go into quarantine. It doesn't matter whether he's come into contact with infected birds or not - the US is going to take no chances.
Oliver doesn't need to come into contact with other birds or people to get sick - he could just as easily catch a claw in something and rip it off, have a seizure or have another emergency. Finding emergency vet care is hard enough when you're familiar with the area and the standards of vet care - I can't imagine how stressful it would be if you didn't know the area, didn't know what the appropriate qualifications/registrations were for an avian specialist, and there was a language barrier.
Why not look at California, Florida or other places on the gulf of Mexico? You'll get sunshine and warmth, and you'll be assured that Oliver wouldn't face any barriers to entry into a different country or return to the US. A number of AA members live in Florida so you could even ask them for vet recommendations before you go.
I really do sympathise - I'm going on holiday in a couple of days to Tenerife. It's within the EU, as is the UK (just about!) so Leia would be allowed in with a pet passport. However, she'd have to travel within the hold of the plane for at least four hours (assuming no delays), the apartment my friends and I are renting is not set up for birds, and if the worst happened, I would really struggle to find someone who could treat an injury or illness to the standards of her vet in the UK. She's spending the week with my parents who will spoil her rotten. She will sulk a little when I return and she's definitely put me in time out before, but I'd rather she was alive and well to sulk at me than the alternative.