Welcome to the club! I saw the title of this thread and smiled, as I just had my braces removed last week and entered my final stages of treatment following an orthognathic procedure in 2019! I chose to work through the process after I commissioned, although I have a classmate who had his treatment done while at USCGA.
Like you, I was aware from the time I was 16 or so that I was going to have to undergo double-jaw surgery to correct my open bite. Medically, I was fine and the open bite never came up during my DoDMERB examination. It did come up during my commissioning physical but my assigned dentist only mentioned that I need to look at surgery in the future. That future arrived when I transferred off my ship to my second unit (a land billet) and had a much lesser chance of deploying. Now, like I said, medically I was fine at that time. However, not correcting these issues now can have long-term impacts - jaw issues, cracked teeth, migraines, etc.
Like the AF, the procedure is considered elective in the Coast Guard so I had to route for approval to commence treatment. That being said, none of my doctors or dentists ever saw my case as elective regardless of what the manual said. The most important thing for my treatment was that I was located in an area that had a Military Treatment Facility (MTF) with an orthodontics section and a oral maxillofacial surgery department. If you're looking to get it done while you're in whatever branch you choose, I would highly recommend you look at jobs in locations that have these services and are also beneficial to your career. It's much easier than trying to route to have it done through a civilian care provider while also worrying about how taking care of yourself will affect your career. For me, the two lined up perfectly. So I routed for approval, got it, and commenced treatment. You need orthodontics first, regardless of if you've had them before, because they need to move your teeth into the best position for the surgery so that your teeth line up when they realign your jaw. I had braces for about two years before I finally had the surgery (a 7.5 hour procedure).
Twelve weeks is the "official" time they'll give you because that's how long it takes for the bones to mend. However, the recovery is far longer. The "initial recovery period" is 6 weeks where you wear a splint, have your mouth rubber banded shut, can't brush your teeth, and run everything you eat through a blender first. It is not pleasant at all, although there's very little pain since the surgery kills all the nerves in the lower half of your face. That being said, it's about three months before you can eat most solid foods because you're reteaching yourself how to chew and using muscles that have atrophied while you recovered, about six months before most of the swelling settles, and the one-year mark is basically what you'll look like for the rest of your life. I was on medical leave for two weeks and took a third week of personal leave just to give myself some extra time before going back to work.
Anyway, that was a long post to say that it is possible to do it through the military after commissioning via a SA and I'm glad I waited. It would have been much more difficult to manage while at USCGA and my unit's Command was extremely supportive of me throughout the process. I wholeheartedly promise you that the process (which is going on 3.5 years for me) is worth the results.