I had researched it once recently for a friend’s dog, and though I can’t answer your question, I would suggest that you email via the Contact section of several of those websites to ask them directly. I emailed one site with another question on behalf of my friend (she can’t use a computer) and I received a complete answer very soon from that site. So I would shoot off some emails and request that if there’s a form to fill out, would they please attach a pdf, and let you know costs.
You probably need a confirmation statement of some kind from your psychiatrist, but again, they should tell you, or you can ask them first.
Btw, I always had very good luck renting apartments. I found that unless it specifically said “no pets”, that even apartments that had a “no cats or dogs” policy would rent to me, and didn’t charge me any fee. The secret is to say your bunny is housed in an enclosed habitat, or say cage, is litter box trained, and makes no noise. The issue for landlords is really damage to their property and noise that bothers neighbors. Bunnies don’t need to be walked, either. So if you don’t mention that you let your bunny out to play, and that they have a taste for carpets, you should be fine. I’ve had 3 apt. landlords write my bunny into the lease before I signed, and was all set. I even got my security deposits back. Don’t limit yourself.