Can a married SM be allowed to stay in the barracks?

1. if you already own a home and you're not even getting BAH yet, then getting BAH for the home you move into on or near post shouldn't be any more of a financial problem than your situation now.
2. if you are married, the Army actually requires you to register your dependents in DEERS as soon as possible, you are responsible for their housing, healthcare and well-being.
3. your wife should be applying for a green card based on being married to a US citizen while on an F1 visa, this is called an adjustment of status:

https://www.boundless.com/immigration-resources/how-to-change-status-from-f-1-visa-to-green-card-through-marriage/

If you're married, you have to register her in DEERS as a dependent. Your wife's legal status to be in the US is something you should have thought about before you got married - as in probably waiting until she got legal residency at the very least.

This might help for getting legal permanent residency: https://home.army.mil/bliss/index.php/about/news/marriage-foreign-spouse-procedure-and-potential-ramifications#:~:text=If%20approved%2C%20the%20foreign%20national,of%20the%20U.S.%20Armed%20Forces.

How is it a financial hardship if you're going to get BAH at your location? That should cover the house where you live. I mean it doesn't really matter if you want to live in an empty house or not. You are married, you have a dependent and if they need rooms for single soldiers, they'll make you move out since you collect BAH. You should have thought this through and planned it out better before making such a big decision. You aren't the first nor will you be the last who gets married and your spouse is in another location, regardless of legal status.

It sounds like at least two issues here. But not having the necessary documentation to register her in DEERs and the reasons why your wife "can’t technically and legally" live with you seem to underpin most of your issues. The advice from this board can be helpful but if you were my troop I'd have you visit with the Legal Assistance Office at your JAG. They also provide Family Legal Services. Within the JAG network, they do provide optional support to non-Servicemember immigration/naturalization issues. In any case, you should check with them and discuss the details of your situation. You need to navigate from a known point and right now it sounds like you could use some legal direction.