E5 new to the unit without dependent approved BAH, E3 with dependent disapproved. Can they do that?

I would recommend you talk to your post finance and have them explain to you BAH type 1 and 2.

1st, do you have physical custody or custodial rights? 2nd, where are you residing? If you are living in the barracks, then you can get BAH Diff only if you are paying child support and you have the documentation to prove it. You reside out of in town and don't have physical custody and pay child support, then you can get BAH with dependents for child support purposes only.

Review regs for BAH, and make sure you qualify. If you are a single parent you must have custody for the majority of the year, documented, to receive BAH. There are also rules on for those without dependents. Make sure you know the Rev before doing more. Going to your Sgt Major is a good idea if you want guidance, but don't go to him if your going to wine.

Question: We're you married at your time of induction? If no, did you demonstrate that you were not the guardian of a child. If you did, that child is not the responsibility of the Army. I once chaptered a soldier out for fraudulent enlistment who complained to our Div CDR that she couldn't get service for her special needs kid. She had assigned guardianship of the child to a relative and after training tried to bring the kid back as her dependent.

It was not your chain of commandbwho turned down your request but Finance or DFAS. Finance qualifies service member's entitlements and benefits. Even if you have a child if you don't have physical custody then you can't have a full BAH. You may want to apply for BAQ-Diff if you don't have physical custody of your child but you have to make sure that your child support is equal or higher than your BAQ-Diff.
Don't compare yourself with another SM. You don't know their situations.

Before you go running to the CSM, have you discussed this with your Squad Leader, Platoon Sergeant or First Sergeant to find out exactly why this was denied? You can't just jump to the top of the chain of command because you don't like the answer or don't understand something. If that was the case, why stop at the CSM, just go see the installation commander, or on your four day weekend take a drive up to Washington DC and ask to speak with the Commander in Chief? Start at the lowest level and ask for an honest explanation about why you were denied BAH. That means they need to get with your company operations NCO and make you an Appointment with the S1. If they can't give you an answer then you let your PSG and 1SG know that you would like to take this matter higher and would like to speak with the CSM under his "Open Door Policy". If you jump the chain of command they can legally punish you under Article 15 of the UCMJ.

SGM Marquez is right. I'm Navy, but both times I reported overseas (as an E-5), because all the E-4 and below were filling up the barracks, there wasn't room for me, and all I needed was an SNA/LNA(Statement/Letter of Non-Availability, depending on whom you talk to) from the BEQ manager, and my request for OHA (Overseas Housing Allowance - almost the same as BAH) was automatically approved. It's also a DoD-wide policy that anyone with dependents listed on their service record (Page 2 for Navy) is automatically entitled to BAH since a BEQ/BOQ is not a suitable location for co-habitation of any dependents, especially children.

It's probably because you do not have full custody

Without shared or sole custody you are only authorized BAH-diff. But you have to be paying child support to get it.

"BAH-Diff is the housing allowance amount for a member who is assigned to single-type quarters and who is authorized a basic allowance for housing solely by reason of the member's payment of child support. A member is not entitled to BAH-Diff if the monthly rate of that child support is less than the BAH-Diff."

It is based on rank. Most bases have a policy for E-5s and above, and E-4s and below (E-6s on some installations). If you don't have more than 50% (by court order) custody AND the child resides with you more than 90 days consecutively, you do not get BAH. Rank is exactly the basis for this, as you have not earned the pay and responsibility of living off base unless you have a family that lives with you.