I would leave it a little while.
Doctors will automatically refer to a clinic and that's not really a route you want to go down at the moment.
Support her as best you can, there is support online but be careful where you go for it. Mermaids is not a place I would recommend.
Speak to the school and see if they have an LGBT club that your child could attend.
Personally I held off on calling my child by their preferred pronoun and name for around 18 months, I wish I hadn't, because it really wasn't such a big deal to call my child by a different name when i think of the bigger picture.
I looked at a lot of the detransitioning people on you tube, and have looked in detail with my child about the surgeries (we watched videos online) and have read up a lot on the negative effects of hormone treatments etc.
After a good few years of my child feeling this way (around 7 or 8) and years of my child presenting as a member of the opposite sex (5 or 6 years) they seem to not be as sure any more and we are possibly lookimg at a future without hormones and surgery now so try not to worry too much.
I managed to find a private counsellor, who is gay and have the same biology as my child and that has been amazing. It was hard to find a counsellor who is supportive but not encouraging but they have been worth their weight in gold to us.
Also I got a lot of support on MN as well. The feminist board was so helpful to me.
I would hold off talking to her dad at the moment too tbh, wait until she wants to tell him or until she wants to push further with a name change or something.
Sometimes these kids can go through a phase and then feel like they have to continue because so many people know and they feel daft changing their mind again so I really wouldn't make a huge deal at the minute.
You can PM me if you would like to chat more, it's a hugely difficult situation and very hard to navigate through.