I’m sorry you’re having a hard time finding an electrologist! That’s really rough.
In my experience, most (if not all) professional electrolysis providers are going to require some identifying information on a damage waiver for liability reasons. Basically, we are asking you to sign a legally binding document that says you understand the risks associated with electrolysis and that you consent to having the procedure done and that you won’t sue if you don’t like the results. Asking for a legal name is just to have a record of who the person is that’s receiving the treatment.
All that said, you should definitely be able to find someone who can treat you respectfully and tactfully with regards to your legal name. I know Louisiana is pretty sparse on providers, but you should absolutely demand and expect someone to be baseline respectful of you. Don’t know what part of the state you live in, but if you can travel and have the will to, maybe consider something over state lines. Good luck <3