There's plenty of evidence that mental health issues are more common among the trans population. But I've learned that we need to look beyond that kind of information to see the details. See e.g.
http://www.npr.org/sections/health-...the-complexities-of-transgender-mental-health. E.g. one study showing a large difference involved the following: "At the same time, most of these women were unemployed, more than a quarter lacked health insurance, and many lived in extreme poverty — all mental health stressors and potential effects of discrimination." Presumably this is not going to be the case in the military, assuming the military can stop its own people from bullying them.
The article showed a study of younger children with no problems. But they had good support.
I have limited time for research on this. I looked for good review articles that looked at the causes of mental health issues. This is the main one I found:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4887282/ It shows evidence that key factors are bullying and other attacks vs support. I don't think anyone knows what would happen if we could create a world with complete acceptance, but at least a fair amount of the problems are due to discrimination.
So what's the impact on the military? You could argue that the military isn't a place to help improve our society. But like it or not, it is a microcosm of our culture. Trying to remove any class of people who have potential issues would involve not just sexual identity but income, education, marital problems, and all sorts of other things. Furthermore, it's unlikely to succeed. So I think we'd be better off to make the military one place where you're accepted based on your ability to do the mission, and where discrimination due to other things is dealt with. Historically, minorities have found the military a good place to advance, precisely because it did accept anyone who was willing to do the mission.