I often say that a setting is define just as much by what it doesn't have as what it does have. I know that for many, the subdued fantasy themes of Birthright was what make this setting attractive; the last thing I'd want out of it is another "everything goes!" setting. I know that for some it's "another humans, elves, dwarves and halfling setting? yawn". For me it's "another cantina bar setting? yawn". I like a tighter focus. This focus doesn't have to be on human/elf/dwarf/halfling, but this is what was established in this setting and diverting from it would lose much of it's attraction in my case.This is whhy I disagree with @Laurefindel 's proposal to keep it as European as possible. I think a lot of newer players (who are now the majority) would find that rather limiting. If it's a Dungeons and Dragons (tm) setting, there should be ways to play all official races. Including rabbit-people.
That being said, a subdued medieval europe theme and high fantasy races aren't completely incompatible in Birthright. "Monsters" in Cerilia are very anthropomorphic in behaviour and often in appearance. Awnsheghlien are a thing, and a rabbitfolk makes a good one. Perhaps your character is not a rabbit-person, he/she is THE rabbit-person! Or perhaps somewhere there is an obscure domain ruled by a king/queen rabbit and its rabbit people. In one way or another, a player could easily make a rabbitfolk character fit in Birthright, only, there won't be many more abroad and the rabbit character will be seen as a curiosity, possibly one that would be persecuted if it weren't under that protection of the local monarch.
I once thought that Birthright's map could even be modular, not unlike some board game where the world/galaxy is generated before each game, allowing the DM to include/exclude/juxtapose/add new domains to fit their Birthright game. I'm no longer sure that's a good idea, but it would allow playgroups to have more control over what they expect out of Birthright.