How can I check the status of my BAH back pay?

You need to go back to your finance office where you're stationed.

This is a tougher challenge than most ppl think. I worked a similar issue for one of my members when I was on AD. It’s a shame to know that when a member is overpaid, the service doesn’t hesitate to reach into the member’s pocket for reimbursement but when he’s underpaid, they’re slow to admit that a mistake was made.
Best resolution practice: have your documentation in order (proof of underpayment). Get your command involved thru the appropriate channels. It takes a pitbull of a Sergeant Major to cut through the red tape while the finance ppl are finger pointing as to whose fault it is for this oversight, all of which, is effectively slowing the process. Be patient. This isn’t the only challenge the command is working at any given time.

First have you brought it up to your team leader,squad leader, PSG? If you have and the haven’t done anything to help request to speak with the 1SG. They should be helping to get you squared away. But yes your S-1 should have submitted a pay inquiry and if needed they would have sent you to Finance or taken you to finance.