Hey! Wanted to let you know that I was able to find a fix for this. I’ll be making a main post on here in a few so more can see.
To begin the “Basic Training” portion of the quest you must be wearing a fatigues and an army helmet. There is an optional objective to search the Barracks for Pvt. Archibald’s Uniform Voucher.
When I first completed the quest, I didn’t look for the voucher, and wore the “Military Fatigues” and “Army Helmet” which I already had. The Master Sergeant npc accepted those, and I was able to start the training, but encountered the bug I mentioned.
Today, I decided to try looking for the optional objective. Using the voucher gives you “Dirty Army Fatigues” and “Dirty Army Helmet”.
I then completed the quest again wearing these items, and it worked! The quest completed. So the issue may be related to the different types of army fatigues