Since you are a voluntary geographical bachelor, you are not entitled to reside in military housing, barracks or bachelor housing. That is established in the Joint Federal Travel Regulation (JFTR). If the you desire to live in the barracks, you may have the option to voluntarily give up BAH, and live in the barracks. I know this puts a financial hit on your family income, but you can’t legal have both.
Think of it like this: A Soldier lives in family housing with his dependents (not drawing BAH). His dependents move out for whatever reason. Now, he wants to remain in housing, but start drawing BAH to support his dependents. It is not legal. Basically the same standard applies to your situation.
You are not required to pay all of your BAH to his dependents in NYC. AR 608-99 governs support of dependents who are not living with the Soldier. By this regulation, depending on his rank, (see attached) you have to pay the amount in the BAH RC/T with dependent rate column for your rank to your dependents. You get to keep the remainder of of your BAH. This applies as long as there is not a court order establishing an amount of support or a signed agreement between the Soldier and their spouse on an amount of support. Either of these two circumstances takes precedence over the chart amount, whether it be higher or lower than the chart amount.