Hi! Hope you don't mind me putting my two cents worth in!
I have Managed swimming pools for 11 years. There is no policy that states goggles are not to be worn. The only policy there is, is that snorkles cannot be worn (unless in proffesional sessions).
If I was you, I would ask the swimming pool's duty manager and ask if there is any such policy regarding goggles. If this was the case, goggles wouldn't be allowed to be used during regular public sessions.
The only risk to your child is if your DD gets kicked in the face by another child. However, as long as the group she is in is taught in an organised manner, the risk is extremely low.
I have been kicked in the face a few times, and the worst i got was a light bruise.
If you went to ask your doctor for a note, they would for a start look at you in disbelief and another they would probably charge you for the pleasure!
I note from you should suffice.
Sounds to me that the teacher is just being a pain in the a*!!
Good luck and stick to your guns!