As stated above, this is a privacy issue. As Dan stated, things were different in the past (let's just say before Truth in Lending). In fact when my father was enlisted in the AF in 1954 he was told by his commanding officer that there would be no loans from banks without his approval. So to buy his 1st car, he had to get his boss' approval. This was probably military custom and as I recall for anyone E4 or below. The military's position back then was that their members paid their debts - even 1-stripers.
Fast Forward to today. I retired from the AF in 2004 as a 20-yr officer. It is still the military's position that ALL their members pay their debts. But I don't know that a bank can simply call a Unit Commander. As RR Sarah points out the Army has some guidance (for it's members), and it alludes to the fact that Reg Z has some information (I didn't read it) on this. If nothing else, I would call the local base's JAG and talk to one of their lawyers about what banks have done in this situation, being mindful of the privacy issue. But I would read Reg Z first and talk to your own lawyer. If you ever do talk to a "commander", make sure it's a Unit commander. The 3-Star has better things to do, than take a call about a bad debt. So it would be good to know the unit this person belongs to. You would most likely be talking to a Lt. Colonel (AF, Army), or a Commander (Navy).
It was my experience within the officer ranks in a squadron, that if a person had a bad debt, the entire squadron knew about it (their credibility would be zero). Thus it was rare to see. I'd bet the mere mention of possibly calling the Commander would go along way toward resolution.