Is this whole issue potentially negated by establishing (as someone suggested earlier) a schedule of damage fees which would be applicable on cessation of the lease?
- Urinated carpets? Replacement carpet dollar cost.
- Holes in walls? Replacement sheetrock, rendering, painting and labor cost.
etc etc.
Provided the schedule is established in advance, regardless of the presence of pets, this should be enforceable as a legal civil contract between you the landlord and the renter(s). They can have their ESA, but it's going to cost them if there's damage to the property.
Regarding landlord insurance, if the presence of any animal invalidates your existing insurance, it may be worth seeing whether this change in cost to you, can be passed through to the tenant, as it constitutes a direct and attributable loss to you, by accepting them.
Disclosure: The above are ideas; I am no lawyer so the above should be validated by one should you wish to adopt such a strategy.