Assuming the U.S. military has already solved the problem of integrating women into the armed forces (bit odd that this thread has focused so much on that issue) then what new problems arise from allowing transgender folks to serve openly as such?
Sometimes when threads wander a bit, it's best to try and understand how we got here.
My premise has been that we should be cautious about introducing change into the military. We should see what has worked in other modern armies. We should pay attention to problems, if any. We shouldn't proceed based on some egalitarian notion of what ought to be, but we should be careful and above all, our only concern should be whether it makes the military better, or worse.
(Thanks to tubbablubba for pointing out that "better" is not exclusively related to killing ability. On the other hand, that whole killing ability thing really is a big deal.)
My discussion about women in the military was mostly based on a question from a user about whether we had done scientific studies of straight, white, male armies. I noted we had thousands of years of data on that subject, and that led to the questions related to suitability of women in pre-modern armies.
So, back to your question.
My answer is that I don't know, but it's a question worth asking, and proceeding cautiously. From my little bit of reading, I note that the US doesn't accept pre-operative transgenders as new recruits, and the UK discourages them. (I don't know exactly what "discourage" means in this context.) Both militaries seem to define "transgender" as including only those people who are actively participating in a medical program that will result in gender reassignment surgery, or people who have already undergone such procedures. If you "self identify" as a woman, but have male equipment, you're going to be sleeping with the guys.
The one thing I did note in my reading was that at least in the UK military, transgenders undergoing hormone therapy were prohibited from certain deployments. That means, to me, that there is at least a small penalty for including transgenders in the ranks. Suddenly, someone participating in a voluntary program can get out of a deployment. That seems a bit strange in a military perspective. However, the modern military isn't John Wayne and the Marines preparing to land at Iwo Jima. Things are different today, so maybe it isn't a big deal. I'll trust the professionals, which includes people whose lives literally depend on the outcome, to make the decision.