This probably won't help too much because I'm in the Navy(E-7), our (the Navy's) pet regs are a little bit more relaxed than the Army. We can't have anymore than 2 pets in any military housing. There are very few Navy housing sites that do not allow any pets, some have size restrictions (less than 25lbs), and some have breed restrictions. IMO I don't think it is a good idea for a dual military couple to have any pets, what if they are both deployed? I know there is a way to take pets on Space-A (military) Flights, but it has to be in conjunction with PCS (transfer) orders and you have to travel with the pet. There is more detail of course but I'm not too familiar with it.
As far as the pet policy in Navy housing in Hawaii which I am intimately familiar with, we are allowed to have pets, the only restriction being Pit Bulls and Rotts are not allowed. However, getting your pets into Hawaii is a MAJOR pain unless you want them to be in quarantine for 4 months and pay about $1500 a piece for said quarantine not including shipping them out there. And the Navy will only recoup up to $500 per pet. There is a way to get around the quarantine but I won't get into that here. If anyone is interested just google "hawaii pet quarantine".