There are a ton of complexities here related to why insurance for carrier capacity is becoming problematic. 

If you can't find a local agent to help, I'm licensed in 45 states and have current clients in Texas and would be happy to help or help point you in the right direction.Any agent should be able to provide you with Evidence of Insurance upon request - it shouldn't matter if it's with the same carrier as the HOA.

Aside from that, I recommend you establish an Assistance Animal Policy that provides additional reasonable elements that must be followed, such as requiring a current veterinarian's report; requiring spay/neuter unless contra-indicated by the veterinarian; identifying chip or collar; and, in the case of a Multifamily property, animal must be carried or in a carrier when in confined common areas such as elevators and stairwells, unless doing so interferes with the animal's task or service for the disabled person.

It's possible that your tenant said Additional Insured and the insurance carrier might need more info for that.

Know that premium brands like Carrier use a lot of parts in their units that are Carrier specific… meaning when a part fails it has to be replaced with a Carrier part - not a generic that a repairman would have on his truck.   

I send oversize postcards to an entire carrier routes.

Hi Errol, To your point, carriers will not insure in certain counties or zip codes if they’ve reached “capacity”.

Having water damage in your three year history of the house reduces the number of carriers, this will fall off as the claims age.

@Jehu Matthews welcome to BP; you found the right place to start on your real estate investment carrier!

While most insurance carriers will likely maintain existing business if it is performing well, they are currently accepting new business at significantly higher rates.