Is her verbal skills really good? I mean, is she able to have a good conversation with you. Do you understand her when she tries to explain something to you, and then vice versa, when you explain something to her?
You could try role-playing... lots of it. Is she able to tell you what happened at school? Then role-play what alternative things she could do/say instead of biting.
Then role-play other situations (that you think can happen in school and other playdates) and give her more ideas of what she could say/do to play with other children. Show her what typical things other children might do, and give her options as to what she can choose to do in return. Exaggerate your happiness and praise when she picks/role-plays a better choice, etc....
As her intensities will continue, you could also try working on a "scale" with her. Like, if all her toys were gone and we can't buy any more, that's a 10! But if the milk spilled, that's a 1. And maybe if you fall off your bike, that's a 4. So, when something happens, let her decide what level on the scale it is.
We also spent a lot of time with our own child to work on perfectionism as that is a typical problem with gifted kids. So we role-played a lot and we exaggerated our own 'forgiveness' when we made mistakes so child could see how we reacted when something didn't go quite right.
It is great that she has an understanding teacher!
