I'd suggest that the red tail WILL nip fins when he gets larger.
They start out rather peaceful when small, but as they approach six inches, they can become very aggressive. The betta most likely won't be fast enough to avoid fin nipping.
The betta should be able to stand his own against EITHER the tigers or sharks, but I wouldn't put him through it.
We have noticed the aggressive behavior coming out in our rainbow sharks. They are now tormenting our Tiger barbs! I assume that the temperament of the rainbow is similar to the red tail...