I'm so sorry you're going through this.
Although your daughter hit her head pretty hard, it is a very unusual pattern for a brain injury to take 5 days to result in headache and vomiting, so it seems like the fall may not have caused her symptoms especially since the CT scan showed no brain injury. She "cried a lot but seemed fine" is usually a clear indication that she was indeed "fine". Kids do bang themselves up and recover in a remarkable way. If you said a few hours after the fall she was vomiting with a headache, that's a completely different story.
I'm confused about your question about whether a 3 year old could damage her vertebra, and the doctors saying no. Obviously, a 3 year old CAN damage her vertebra - that's beyond debate. I wonder if your question was can a 3 year old suffer vertebra damage from a fall where there seemed to be no evidence of a head injury and symptoms didn't appear until 5 days after the event? Was your question actually, "could my child have suffered vertebra damage from the pool incident"?
I think you need to look elsewhere - besides the fall at the pool - to explain her back pain and loss of toilet trained behavior.
I do sympathize with your anger that you weren't allowed to supervise your child at the pool, and she was allowed to run on a wet surface.