Your question isn’t one that has a single, universal answer, but I will try my best to give you helpful pointers. I will also have to guess at some information I don’t have or nuance my answer depending on what your precise situation is.
I understand your worry that your dad might hate you if you come out. It’s a perfectly normal thing to be concerned about, at any age, and as kids of course, we want our parents to love and respect us. That said, your dad’s possible reaction should not factor in with your decision to be who you are as a person. You seem to be on the right path already, saying you’ve decided to begin transitioning when you’re with your mom, which indicates to me that you aren’t waiting for his approval or validation. Just remember that your concern about him hating you might still affect you in other subtle ways. Understanding this is the best way to deal with that.
There’s no sure way to know or predict how someone will react to a coming out, but him seeming transphobic makes it likely that he will react negatively. Transphobia is not a rational thing. It’s based in ignorance, misinformation and fear of things that one does not understand. It also has nothing to do with you as a person and has everything to do with his choices (hating is a choice). It’s his own responsibility. I hope, for your sake, that if you do come out to him he will choose to change his thinking, but there’s no way to know how things will happen.
Knowing all this, whatever you decide to do, your main concern should be for your own safety. At your age, you’re dependent on adults for essential things such as food and shelter. This means that you have limited options if coming out puts you at risk of physical, psychological or economic violence (generally defined as using control over financial resources against someone). If you think your dad might react or act in a violent manner towards you, it’s best to have the option of not living with him at least available to you, just in case. It could be through waiting to be old enough to leave or through having the option to only stay with your mom (who, I assume with the way your question was formulated, is a safe environment for you right now).
Another possibility is to transition when living with your mom, as you’re thinking of doing, but not tell your dad about your identity just yet. An important thing to know if you do it this way is that, depending on the nature of what you mean by transitioning, he could find out. For example, if you change your clothes and appearance, he or someone else could see you and learn about it. If your identity is known by other people, he might hear about it. And if you use puberty blockers, the lack of puberty might also become noticeable. What I’m saying is that keeping a secret like that might be safer for you for a time, and it might be the best path, but if you choose to do it that way, you have to be aware that the secret might be broken at a time not of your choosing.
Lastly, I want to mention that you should (if you haven’t done so already) try to learn about the laws in your state regarding transgender children, as the laws are very different from state to state and, in some places, they’re very problematic.
I’m sorry that I can’t provide an easy or reassuring answer and I hope things go well for you. You’ve already taken a big step in knowing who you feel you are and deciding to be your true self, and that’s an accomplishment in and of itself.
Sophie (elle/she), volunteer for AlterHéros