Tawny is cute. It isn't a name that I would use, but I like it.
Tawny sounds like a fairy in a children's book or a cute anthropomorphic animal.
The spelling
Tahnee looks like it would be pronounced differently with more stress on "ah." I'm not sure that I understand wanting to name your child tawny but simultaneously wanting it to look as different as possible. Why not use a different name?
Taranee makes me think of tranny.