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

Hard to say without a closer look at your particular situation. If you want clarity, go visit the IG and ask them. They will sort out the truth for you.

was your child born before you joined or while you were in. cause that actually plays a big factor...

PFC Blair, Let me rehash this for a minute,

You are asking if you should get full BAH for your son and be allowed to move off base? However, based on your other answer, you are already receiving BAH Diff, and you are the non custodial parent, is that correct?

If so, based on those facts alone, the commander is the only one who can authorize you to move off base. As a PFC he's very unlikely to do so unless the barracks are full. Unfortunately that's the reality of your situation as a single soldier.

As has been previously stated, if you have primary physical custody of more than 50% of the year or 6 months out of the year, you are then authorized full BAH and have to move out of the barracks. Going to your CSM won't change the policy or your situation for the better based on all the information you have given us.

I'll also caution you about comparing your situation to the SGT's. One yes his rank will play into the decision of housing, as typically NCOs are more frequently given permission to live on the economy over Junior soldiers based on their demonstrated performance.

Well Sgts typically rate BAH to get them out of the barracks. Is your dependent in your military record? Is he actually your son or did you adopt? If he's not actually your kid or you haven't registered him with the personnel to get him added to your military record, then no you wouldn't rate BAH. But if you have him documented, then you should rate BAH. It's not a command decision, it's a personnel decision. You should talk to your personnel department. For Marines it's DEERS/RAPIDS/IPAC. Not sure what it is for Army.

PFC

Too many unknowns to give you the real answer. Do you have custody? Does your unit have you living in the barracks? The Sergeant is another issue that hardly concern you. Be a man and talk to your first line supervisor. If that don't work, tell them you want to speak to the next higher up, etc, etc..... Don't post part of the story and have people try to figure out why. Get all the fact together on your situation then ask your chain of command at the proper level.

The answer is very simple, AR 608-99

Doesn't rallypoint sometimes add pictures to topics based on the subject? I think I've seen several post like this before and saw someone mention this.

MSG (Join to see) change what pic and why??

Do you have custody of your child?

I'm just going to say this:
1) Choose a different pic.
2) Did you bother to ask your Chain of Support if they could explain why before trying to blast the decisions of your Command on this site?

As SGM Marquez said, you need custody of the child. This needs to be a court document. Visitation alone is not enough, it must be custody either, sole, or joint.
Several posts do not have room for E5 to live in the barracks. So, BAH may be authorized for them to reside off post.
If you have joint custody of the child then goto housing. Housing will determine if you are authorized to not live in the barracks based off of DEERS, and custody agreements.