First of all heaps of praise for you for your post and bravery. Wow, you made my day posting here on this topic.
I'm so happy your father had embraced your sexuality transformation and in particular using your name and words that endorse your new identity. You are very lucky.
My daughter came to live with me at 12yo and her mother used the victim approach with that. I explained to her that it is HER life and her choices. When her mother was her age at 12yo she made her own choices also. You would ideally be better off in a home that gives you full support during this period of the next several years otherwise any negative responses will drag you down.
Leaving home to live with dad can be expressed positively towards your mum by often telling her you love her and that you are happy. If you make the move you will have less time with your mum and that might mean she will appreciate you more even as your new identity.
In defence of your mum I can sympathise with her with not remembering your new name. My Daughter likes to be called (not her real name) say Sam instead of Samantha but I cant ever get used to it. My sister Sandi from Sandra and so on but once you get a few decades along lifes track it is hard to change. So please by all means remind your mum of your new name but please try not to hurt her for not remembering ok. She still loves you.
I hope that helps you take one more step forward. Remember- this is your life, not anyone elses. You make your decisions that will improve your life for the ultimate goal- happiness for being yourself...