In, at, or on

I think you want "in base housing" ("in" and no hyphen).

"on base housing" gets too tangled up with the expression "on-base housing" (with a hyphen!), and if you use that, you'll end up with "in on-base housing", which is really too much of a good thing!

In my opinion, you want to take the hyphen out of "in base-housing", and stop there. The sentence is fine.