I work for the Army as an Education Advisor. I hope I can clear up some confusion.
Whether you are entitled to this or any of the benefits he earned by serving depends on your state's laws. You need to talk to an attorney who is well versed in military divorces. But be aware....no matter what you ask for, that money isn't for you, it's for the care of the children. Be careful how you word this when you address it in court.
You are not necessarily entitled to any of this money. He gets it for going to school to cover school related costs of living, not family related costs of living.
Your husband is using the Chapter 33 Post 9-11 GI Bill. The BAH he gets for going to school is the E-5 rate, with dependents, for his zip code, and based on how many months he has deployed overseas. It has nothing to do with you or the kids. It's just the amount he's entitled to for going to school. I advise that you leave this alone entirely, so as not to seem like the hateful woman who would cause him to have to drop out of school because he can't afford gas to get there. Go after his paycheck, not the VA benefits he EARNED by serving. They are NOT the same thing.
If he were active duty, you'd be entitled to part of his active duty pay-related-BAH because he'd be bringing home a regular paycheck and the BAH with dependents part of his paycheck would be FOR his family.
The BAH he is getting as a part of his VA benefit for going to school is an entirely different animal altogether. It is to defray the costs of going to school.
HOWEVER, regardless of your state's laws, his children should be getting some type of support out of his paycheck.
I wish you the best of luck. If you have any questions, don't hesitate to message me.
ETA: The BAH he gets for school is NOT based on his family situation. Regardless of whether or not he is married with kids, he is using Ch 33 and therefore gets the E-5-with-dependents rate.
I know it's confusing, but education benefits are not the same as regular BAH.
HOWEVER, I do think that he should be sending some money for the care of his children. You need to contact an attorney and get a separation order into place in the interim so that there is a court ordered payment in place.