I have a friend who has a service dog. In order to become a service dog it has had extensive obedience training, a canine good citizenship test and pass a public access test. You may be able to buy a vest anywhere, but service dogs are also required to have the paperwork stating they are a real service dog. While it's true that no one can ask the nature of your disability, they can ask for the paperwork proving the dogs status. My friend's dog is a seizure alert dog (she has epilepsy) and she is constantly asked for her paperwork.
An emotional support dog does not require the training. You have to get doctors orders for the dog. The only reasons a person needs documentation a dog is an emotional support animal is if they require some sort of accommodation, or exception to rules that would ordinarily prohibit pets, for example, an apartment that doesn't allow pets, or riding in the cabin of an aircraft that doesn't allow pets. Those are the only two places where emotional support animals are permitted that regular pets are not. The timing of your paperwork can also be put into question by the landlord. Why was this dog not an ESA before the move...why now suddenly?
I see my friend who needs her dog struggle often. Even with ADA backing, she has been denied entrance to places. It has happened more than once. I implore you not to try to bend the rules to suite your needs. This house may be fabulous is every other way, but it does not meet the needs of your whole family. This is simply not the house for your family.