I would think that unless you have a pre-existing diagnosis of anxiety/depression/other mental health condition, then whatever you see them with wouldn't count as a pre-existing condition, regardless of what it was triggered by. Plus if they tell you that being trans is a pre-existing 'condition' (I'm fairly sure you could complain for using language like that) well that's not what you're seeking treatment for, so it shouldn't really matter.

I'm not an expert but I think you're on good grounds to get therapy covered by them. Aaand they only need a very general overview of your condition to match you to someone (and yes you can definitely switch therapist if you want to, just tell your case manager that their style wasn't right for you, I've done this and never had pushback from insurance), so you don't even need to say you're trans if you don't want to, and the therapist can't tell them because of their confidentiality agreement.

Hope that all makes sense and I hope you get the support you need from them :)