Hey, Brandon. Sounds like you feel stuck and that there is no way around this. I would suggest hoping that things will be different next year (that you can maybe go to prom in a tux with a girl), and just accept that things won't work for this year. I completely understand just simply not being able to work around your mom.
It wouldn't have worked out for me to just say I was going to prom and not have my parents (mom) involved in the proces. This wouldn't be because they are jerks, it is just that they simply wouldn't have understood why they wouldn't be involved in helping me get a dress, pictures with my date, etc.
I think that there is great diversity here on the forum around the level of independence/separateness people have from their parents. When people post about wanting a male haircut for example when their parents won't let them, some guys suggest people just go and do it. That never would have worked in my world. It just wouldn't have. I can't even imagine going around my parents that way back when I was a teen (this was 20+ years ago). I would drink, smoke cigarrettes and stuff like that behind their backs, but I can't imagine just getting a male cut and thumbing my nose up at them.
Anyway, I get how you feel stuck in a lot of different ways due to your circumstances, is all I am saying. I would suggest keep looking and focusing on the future when you are on your own and can do what you want. Don't dwell on what you can't do right now. It is only a couple of years until you can do what you want. Just date the girl on the side and don't meet her parents. Meet her at movies with groups of people. That kind of thing so that you can spend time with her.
Also, I am a bit suprised that your mom doesn't know you like girls. She knows you are FTM right? I am surprised you would tell her that, but not the liking girls part.
Look towards your freedom to be you in the future, and hang in there until then.