So, I ended up calling up various "pet resorts" (I had no idea these places existed) looking for lodging accommodations for his cat.

This is exactly one of my pet peeves!!  

I had already waived the cleaning fee ($200 pet fee) and didn't charge her the $25/month pet fee because she was relocating from out west back to Michigan and money was going to be tight due to the move.She always took care of the property for the 5 months she was there.

"Then they will say location, one bedroom, bath to small, extra pet fees, no garage, rent to high for what it is, etc., etc., etc.

I do not believe that a landlord would be pleased to sit by for weeks or even months in a vacant rental (who knows what is in the fridge or what pets they had, etc) waiting for an estate to get settled.Why not just call the executor, ask what is to be done with the belongings in the rental unit and where to send the deposit check?

First, last, $1K and $250 per pet to move in.

Put lower realistic offers and, YES - practice with the negotiations: as long as they don't accept: you will not lose any money (emd ), everything gets serious after that grace period after an acceptance (before that, I would ask for the pets and furniture if I could).  

That was a niche that got me 30% higher rents than a no pet 3/2. 

We rehabbed a house a couple of years ago with severe pet & smoke odor.

Think of all the things that could go wrong and see if the lease addresses them: unauthorized pets or tenants, early termination, security deposit, lease violations, late rent, eviction, lawn maintenance, parking, etc.5.