Trans dual citizen with two different legal identities

I'm in a bit of a unique situation. I have citizenship for both the US and Hungary, and I have had my name and gender marker changed in the US, but Hungary doesn't allow either. My partner's job is sponsoring a visa for her to move to Ireland, and I plan on coming with her. My understanding is that I'll only be able to use my Hungarian identification for things since it's the one that allows me to live and work there, that that name will be the one that shows up on any identification I get, and that if and when I naturalize I'll need to again use the Hungarian info and then just get my name changed a second time? Is that correct, or is there any possible way for me to use my US name and/or gender for anything along the way?