I am still somewhat hazy on what is proposed, what problem it is theoretically to address and what the potential fallout is to be. The article linked by the OP spends some time discussing title 9 and how it deals with sexism -- unless we are going to go full moron and go with the "dominant group" definitions of things then you can say (or be demonstrated to be acting such that) you are doing or not doing something on the basis of sex and get pinged either way. If it sought to specifically exclude such things from consideration in such cases (excluding people for arbitrary characteristics is bad, let them compete on merit) then I could see a problem but nothing along those lines. Also as we are dealing with government documents and are nominally a technical site then while I am at it
https://web.archive.org/web/20160731010249/https://qntm.org/gay .
I have previously covered my thoughts on the notion of transgenderism but might as well go again. I generally boil it down to if you are not hurting anybody else then I don't give a fuck, I find it utterly bizarre that anybody would give enough of a shit about their gender to want to change it but there are many such things I find bizarre and it does seem to be real with positive outcomes for allowing people to change. If you do want to change on official documents I would expect it to be the sort of thing a medic oversees and signs off on -- pending better medicine (itself a minefield -- what would you do if you could take a pill and then stop giving a fuck about it?) I thoroughly support the hormones, living as* for an extended period, counselling and more (surgery would be optional but certainly enhances a claim) regime. The more genders than atoms in the universe thing I find to be odd -- if you want to define some group of characteristics and give them a proper noun then so be it, the notions of gender seem useful enough without them and not enhanced by them. I should also note at this point the only people I have ever met that I would usefully describe as fluid with respect to time have been people that changed, (occasionally changed back) and then got the types of dementia that see you living in different periods in your life, now this does not preclude it from existing but I am still awaiting evidence of something else. Sadly I have not seen any claimants be tested a la the Russian blue (Russian has two words for blue, or actually two types of blue but if you strain a speaker's mind by getting them to do complex maths it goes back to the one most other languages and physics/biology recognise) or Australian Aboriginal compass (many such individuals, perhaps most, will be able to tell you the compass direction even inside a windowless building after having being taken through a series of turns, again load their minds down with maths and it goes away) and I am not entirely sure how I would construct such a test either. I would also however need to reaffirm that I am categorically opposed to mandating pronouns by law (which would extend to "government funded institutions can not enact such rules") or "hate speech" specifics here -- it might make you a horrible cunt to do such a thing but it is not and should not something you need to be put in bracelets and stuck before the beak for. If you go punch someone because "they is a t****** yo" you have merely punched someone without reason and such a thing should be able to be handled by the regular laws against not punching people, no problem with it being considering at sentencing though.
*itself an interesting conundrum as I would have thought the idea of gender roles would be tricky for such a group.
"The Trump administration has sought to bar transgender people from serving in the military"
Not sure entirely why it is here in this (other than maybe as a type of rabble rousing) but OK. If you can handle the training (which can be role specific* but I see no reason for it to be different for sexes) and the job I have no issue with transgender people playing here, to exclude them would something that needs some other major justification which I can't see happening (I can't see the trait inherently or so probabilistically rendering them unfit or unable). If we are to have such nonsense as gender specific requirements (training itself if it is merited is a different matter) then it could get a bit hazy but that seems handily solved by eliminating such requirements.
*front line combat vs administrative roles sort of thing (you don't need to be able to carry your fellow office worker a given distance). We can discuss resulting efficacy if you like as well, and there is some data there on mixed units.