You have an uphill battle
Take a look at the DoD
FMR (Financial Mangmt Reg) Ch 26
You can see it herehttp://comptroller.defense.gov/fmr/archive/07aarch/07AIC24-03.pdf
If you look at section 2 it lays out how a service member who does not have dependents can receive BAH.
It will not cover you...not as an E3.
As an E4, if you are in "sea duty" you can qualify....but I do not believe that will apply to you
Here is the key provision 260205(B)(2)
Except as provided in subparagraphs 260201.B and C, a member
without dependents in grade E-6 or below who is offered an assignment of adequate government
quarters, or is assigned government quarters but elects not to occupy such quarters and resides in
private quarters at own expense, is considered to be assigned to government quarters and not
entitled to BAH. Therefore, such member is entitled to partial BAH.
So the FMR is not going to support you other than to give you partial BAH which is $7/mo.
You can ask the commander to waive this...but that would have to come from HQDA (so you would need to get the flag officer
to recommend it)...I suspect you may have a tough time with that
Sorry to have to bear bad news
Let me know if you have more questions...happy to help if I can