I would argue that instead of looking at the bodies to compare too much, compare the lenses more. Fuji doesn't really have a fleshed out line of lenses in the telephoto range beyond 200mm. They have a single 100-400. That's it. So unless you're looking to only do close range, short lenses for birds as you mentioned, Fuji doesn't have a lot of options. Sony has more options in the long lens range, including 3rd party native mount lenses from Tamron, etc.
The T3's AF is pretty stellar, with great low light AF acquisition performance.
I don't think they're different enough in terms of overall image quality to really split hairs over.
The lens selection for your intended purposes and the menu/ergonomics and the AF differences will be probably more starkly different than IQ/DR.