There's definitely some overlap sometimes, but I don't think it's so much a cause or effect necessarily. I think bipolar episodes can certainly be triggered or exacerbated by stress from dysphoria though. During my last manic episode 5 years ago, having finally come to the conclusion that I was trans earlier that year, I started dressing and acting more masculine, temporarily dropping my 'girl' act and even made mention of not being a girl to my mother. Thus, years later, while I'm sane and very functional, she still sees my desire to transition as a warning sign of impending psychotic episode, when really it had just been the loss of inhibition that caused my true feelings to be known.
As for transitioning, it certainly would depend on the particular medical practitioners, and also the severity and type of disorder. My bipolar episodes, for example, are easily recognized as I don't get hypomania, I'm either way out in left field and completely dysfunctional (and very obviously so), or 'normal'; I don't exist in a half-manic state, and yet the gender dysphoria persists even as my moods are stable. I think also, if you can convince them that your gender discomfort existed before you showed symptoms of your disorder (in childhood or puberty, for example), it might help your case. I've been in psych hospitals as well, but I'm not sharing that info with my current therapist. That was a long time ago and I've got things under control now, it's really a completely separate issue as far as 'treatment' goes anyways.