Tough call. This is typical of military medicine, and has been forever. They are very good at some things (ie trauma) but often dysfunctional in routine matters.

You are not going to change anything. If you annoy the wrong person, your file might conceivably get lost, misplaced, etc. Not the norm, but as a recruit you have very little recourse or visibility on military administrative matters so if someone were so inclined they could probably get away with it.

In your present position, I would not push the issue too hard on matters of principle or due to the inconvenience with your current employment situation. If you are going to miss your age cutoff, then use any means necessary to get the ball rolling, you don't have anything to lose in that case.

As far as your employer...wait until you have orders in hand before you give any notice at all. Hopefully you will have at least two weeks notice, but if not...oh well, it's the military nothing you can do about it. No future employer would hold that against you if the military called you on short notice. If you need another job, go get one. The fact the you have APPLIED to the military is totally irrelevant, and none of their business. Since getting accepted is by no means a sure thing, you should not hedge your civilian bets in any way until you know for certain that you are in.