Fab reply above! I can't add much more to it, except to say that there is the issue that your daughter will be having periods starting in the next few years...and I'm sure for that reason you are aware that she will need some way to tackle it now. It is an extreme reaction in one so young, and it would be interesting to try and work out the trigger. I've just done a bit of googling on it, and there are some self-help phobia sites (obviously with your daughter being so young it would need to be you implementing the processes).....also....how about looking at the self-help books listed on this site and seeing if there is one that could be applied to a young child? Whilst I do see where your GP is coming from, I think your daughter's reaction is extreme and does affect her life considerably, and has done for an extended period of time, so some therapy for this phobia seems appropriate. Some ideas listed here...
https://www.healthline.com/health/hemophobia#treatment