I wound up coming up with standardized language to put into the lease (which originally just said that service animals would be permitted according to law). I now require any animals brought onto the property for any length of time, even service and emotional support animals, to be registered with petscreening.com or otherwise verified through a third party, and if tenants or their guests are not happy with that, they can send the documentation directly to me (vaccination records for the animal and a doctor's note with medical license and clinician contact information be sent to me). Upon verification, I will then give them written permission to have that specific animal on the property. Petscreening.com seems a bit predatory, but it's sadly necessary to do this because people are trying to pass pets off as service animals. I also made it clear that he is responsible for any damages caused by the dog, and its status as a service/support animal does not exempt him from paying for property damage or cleaning fees for messes that are above the normal wear-and-tear.
I also clarified my guest policy to the applicant, which is that guests may stay a total of 10 nights, and not more than 5 consecutive nights, if they are not on the lease (the lease is for 3.5 months).
I emailed the applicant to give him a heads up about the language in the lease, in case he wanted to find a more dog-friendly property, but he wants to go ahead. I don't have a reason to turn him down, other than the dog, but if it is a real emotional support animal that would be discrimination, so we're proceeding.
I've had such bad experiences in the past with dogs in a different fully furnished short term rental where we originally allowed dogs-- swimming in a decorative pond and disconnecting the filter, killing the koi, chewing up a couch and other soft furnishings, scratching a newly installed hardwood door and ruining the finish, digging giant holes in the yard and landscaping.... thousands of dollars of damage, really, in a very short time, which is why I have a strict no pets policy, and am particularly nervous about dogs on the property.