That seems odd. I wonder if she learned more information about transitioning and testosterone or if she got feedback from a supervisor or if like you said she gets moody. If this is something that you are sure about and it sounds like you are then I think you should try to set aside her opinions and talk to the doctor. She may have some or even a lot of information about transitioning, but she can't prescribe the medication and so probably doesn't really know if you are a candidate or not.

I know very little about the effect that giving testosterone to a person with a mental illness would have, but just by doing a quick google search I found one medical article that said that there was no evidence that it worsened mental health symptoms and a lot of personal stories that said that it improved their mental health a lot, which makes sense. If someone starts to feel like their body is beginning to match their true gender it only seems logical that they would start to feel more comfortable with themselves and likely experience less depression.

As far as binge eating disorder, do you actually binge? Or do you just eat more than you need to eat? If this is going to stop you from transitioning in some way, like you need some sort of mental health evaluation, I would get a second opinion. This really could be her problem.