Yeah, pets were beyond a cluster during the evac. I stood in line for 3 hours to get someone else's dog seen by the vet to get his evac certificate. It was insane. I was grateful not to have a pet, even though I do really want a dog.
But yes, you can bring them.
Also, I don't know what you fly or what base you might go to, but for Atsugi, people can definitely still live off base. Many do, myself included, and many have pets. A cat will somewhat limit your choices, but that's really no different than trying to rent in the states. I know people with cats and dogs (including a lab, so not just little ankle biters) that found houses offbase with no problems.
Having a pet also limits which houses on base you can live in. This is where it can be tricky. They will offer you the first house (in your alloted bedroom size) available. If it isn't pet freindly and you have a pet, you have to turn it down. However, at that point, they have met your obligation (which is to provide a house for the people, and not to accomidate your choice to have a pet) and if you end up being in the Lodge more than 30 days before they offer you a place with does allow a pet, you most likely will lose your TLA (or whatever allowance it is that pays back your hotel costs?) and be out of pocket until housing has a pet friendly place for you.
I don't know how many places restrict cats vs. dogs, so I can't say how much of an issue it might be, but I thought I'd throw it out there as another factor to consider.