You're actually right and wrong. If you stopped someone from renting because they had an ESA, that's discrimination. So long as there is proper documentation, they can be housed in any (proper) housing with their owners. So long as the owner can own the animal (so no exotics/etc), and the animal is being taken care of to animal welfare standards/not being destructive, you CANNOT discriminate against someone for living with their ESA. That would be illegal. I mean, you can. And clearly you said you would - but in the eyes of the law, it is illegal. Birds are domestic animals, and are allowed to be kept indoors. Even in your rental.
You are right that you (probably) could not bring a chicken in a grocery store, though! ESAs do not have public access - the only animals that currently have true public access are dogs and miniature horses. ESAs are support animals, not service animals.
Service animals are the ones allowed public access. The only problem is that the legal definition of service animal is piss poor. Technically there is no legal precedent for types of animals - although dogs and miniature horses are the only ones with actual training organizations
that I know of - and the amount of tasks they are required to perform has no precedent, either. They have to be trained tasks, and most people will tell you at least two, that relate to supporting/aiding the owner in some way. I mean, if you could train a chicken to perform at least two tasks that relate to PTSD aid, technically it would be legal. However, there arises the issue that any animal considered a threat to public health/safety/a disturbance in general can be asked to leave - whether you have a thousands of dollar seeing eye dog who spent years in training or an ESA just meant to block/comfort, if they act up in any public location the owner can ask you to leave.
Don't let anyone fool you, OP. There are NO certifications for ESAs - all those registries for dogs and such are scams. All that is required, technically, is a doctor's order for the animal. Be aware that you still have to comply with the ordinances set forth by your community, though. If chickens are zoned as livestock specifically, that's a no go for housing/transport. If they allow "birds," or fowl as indoor pets, though, you should be in the clear.
Don't waste your money trying to go through a fake registry. They have no legal standing, and giving them money just gives them more power.
DO be aware that transportation services are not REQUIRED to make exceptions for unusual animals, domestic or otherwise. If you are doing this to try to get the bird on a plane or whatever, as some people like to try to do, it's not going to fly (no pun intended). If Airline A doesn't ship birds and fowl at all, they are not required to let you bring your bird on board. If Airline B
does ship birds, it's a little more fuzzy - I don't know of any precedents concerning it. I don't know if the FAA would allow them to consider it an "unusual animal," if they regularly transport them anyway.
Sorry for the long rant-y post. I have pretty bad anxiety/depression and have an ESA myself, who is just a cat. It's still a pain in the butt trying to do anything with him, though. No, he doesn't have public access, but he is allowed in homes with me. I try to be very up front about the fact that I have him, that he is legal and has doctor documentation, and I've never even had a reply to any housing application I've put in. I can't imagine how difficult it would be with a chicken. Dogs and miniature horses just don't fly for everyone, though. Sometimes you need support from a different kind of animal, and that's okay - so long as you're not trying to circumvent the law of the land and your animal isn't a nuisance
When you have time you can try sprouting some of those to make the nutrients even more bio available. It's easy to do and can be a lot of fun and the birds enjoy it a lot. If you grow the sprouts out for 7 days they even turn green, for a little more enrichment. I know you said you take trips outdoors, you think August gets plenty of grit for digestion when out there? Also sounds like you care and try really hard. 
