"Payment of BAH when member resides separately from
dependents. Payment of BAH based on dependent location for a
member who lives separately from dependents due to military
necessity may only be authorized when the member is not able to
commute daily between the duty station location and the
dependents' residence. Assigned military duties, including
deployment and transportation interruptions, do not constitute
an inability to commute."
From OPNAV 7220.12 Chapter 1.
I'm not sure about the other branches, but Navy and Army have similar BAH guidelines regarding dependent location. Navy generally allows you to choose, so ends up the higher of the two. I collected San Diego BAH for three years while my wife was in Illinois. It's about a $500/mo difference.
Before you jump on me for leaving something out... I'm currently serving with several others from SD who left their dependents behind while executing accompanied orders. They still collect SD BAH while living in Chicago.
All it takes is an approved request to collect BAH based on dependent zip, routed through your CoC. They don't have to sign off on it, but if you're maintaining a residence which costs more than what you're getting it would become a quality of life issue. I've not met anyone who would place their subordinates under a financial burden when they have the ability to avoid it.
Your pay office is who you should be talking to about this. IMO, your 1SGT should have sent you there immediately to get this sorted.