My counter to that Ele is we need to consider why transgender people kill themselves. Shot in the dark - I'd imagine they can't get their gender reassignment surgery, and are generally treated as 2nd class citizens.
The military already pays for ancillary surgeries with a certain amount of commitment (I think it's 3 years?), and well, it doesn't help someone's cause when their country says you're not good enough to defend our freedoms.
Gender reassignment surgery being paid for by the military is a difficult prospect, but not unobtainable. If we're going to spend as much as we do on defense spending, might as well take a moral high ground on a subject like this. From what I've been told, you can already get breast augmentation on the military dime.
Where you and I both agree is the government simply doesn't do enough in clinical/mental health for our prospective and current troops.