I've stayed both on and off base at LR. The last time I went, I stayed in the "new" billeting. It sucks. They can "add" on a room and act like it's great all they want too But it's still nothing more than making a turd into a bigger turd.
The billeting is fine for just one person. But believe me that you don't want to even try it if you both are going.
There are plenty of places to stay in LR that allow pets. Sometimes it can get pricey, but start using some internet apartment finders.
Regarding base housing, don't expect it. Since you're not TDY enroute, losing your BAH isn't an issue. If you are getting BAH from Kirtland, you might keep it. I'm not sure what your housing situation is at Kirtland.
For good info on pay at LR, check out the LRAFB schoolhouse guide.
The other thing to remember is if your husband isn't going to slicks, he won't be at LR very long. He'll most likely only go through phase one and two.
You could consider staying at Kirtland and sending him to LR by himself. Then you could keep the dogs and minimize the problems. The downside being you'll be apart for a while. But you're about to start being away from each other plenty, so it might be good practice for you and him.
HD