D.P.
HI K.,
Can you encourage her to use her words for other things and situations like a drink, snack, bath, toys etc? Also, have you thought about teaching her some basic sign language for common emotions so she could try to express herself that way?
I think you are doing a good thing by giving a time out in those situations. It will allow her to calm down and think about what she has done. You might want to tell her before she is in the time out that "it's OK to feel angry or frustrated but it's not OK to bite/push/shove." Can she apologize after the time out is over? Encourage her to say "I'm sorry for biting" at the end of her 3 minutes. I would also encourage her to apologize to her brother as well.
Good luck! Hope this helps