Can my husband live in the barracks while we are married?

You're listed as a civilian so I'm guessing your husband is the military member. Your husband will be a geographical bachelor, and if he's a SSG or below, he likely will live in the barracks, at the direction of his 1SG.

He could claim geographical bachelor to his command and live in the barracks till this is settled

I had to an exception to policy memo for a married soldier to stay in the barracks. It was conditional. And needed justification. Cant take away from the minimum space required of a single soldier and if they get full theyre the first to go

His Commander will make that determination. If room exists in the barracks, the Commander can approve his living in the barracks. It's not a definite though. Commander's discretion.

You might want to ask your hubby to discuss this with the 1st shirt/Cdr. back in the day, it was permissible, unsure as to now, if he does stay in the barrack, he may only be entitled to separate rations entitlements vs. the family BAQ entitlement. While in the military, I've worked under the old regulations, because I use to do this type of paperwork for soldiers. So much as changed over the years, it's best for your husband to converse with his Chain of Command about this matter, or, he can find out what the rules are online under the regulations. Hope this helps out :-) .

The Army got rid of the geographical bachelor program. Your husband will have to rent a place on his own, probably rooming with other single soldiers.
https://www.thebalancecareers.com/army-announces-end-to-geographic-bachelors-3353934