Can an Emotional Support Animal live on on-base military housing if they are named on the banned dog breed list?

Here's what I found:

"While only dogs are considered service animals under the ADA (and some accommodations may be required for the use of some miniature horses), the Fair Housing Act has no such restriction. While the ADA does not include emotional support animals, the FHA does. Some housing providers will be subject to the ADA, such as government owned housing. Some will be subject to the FHA, such as most landlords with more than four units. Some will be subject to both, and some to neither. Wherever the FHA applies, common domestic species are included as assistance animals, but where the ADA applies, only dogs are. Where both apply the FHA multiple species policy applies."

It appears that government housing is not required to provide accommodations for emotional support animals, but they may be willing to make an exception.

Source:
https://usaservicedogregistration.com/housing-laws-service-emotional-support-animals/

Of course a common loophole I've heard of is people just saying their dog is a mutt, not a purebred.

You have received sone great advice. Follow the policy. One thing you left open was who’s dog it is. It cannot be for both of you. As for a Soldier he cannot bring an ESD to work, training, or combat.

An exception to policy may require medical verification. Just like when flying the ADA requires the psychiatrists general statement that you are in need of a service dogs support. It does not account for most airlines regarding ESD. There are some confusing language associated with the difference and training required. It’s muddy don’t wait and don’t lie about the breed as you may be evicted once found out.

Military housing is further complicated because there are two separate set of rules.

If you are reassigned later as a Family to Germany you would not be allowed the dog period based on the breed. This is a broader issue than you may realize.

Not trying to give you further concern, just making you aware that it is not as simple as saying you need the ESD, there are broader implication and should be thought through if possible.

Is it your or his.

The Fair Housing Act Emotional Support Animal protects individuals with ESAs. If an animal helps an individual with mental or emotional illness, as per the Fair Housing Act, they can rightly keep their emotional Support Animal in a rental housing Authority. The Emotional Support Animal can be of any breed, size, and weight. you can check more here : https://fastesaletter.com/emotional-support-animals-the-fair-housing-act/