I heard an interview done on them and they had some excellent ideas on how to pitch rental arbitrage to a landlord and how it can benefit them and their property.

This sounds like it might be a win-win, Hospice has plenty of funding to be a reliable renter. 

I am not familiar with the Cleveland, OH market, however, if you do not find what you are looking for and want to look at other excellent investment markets in the South East, then you need to take a look at Birmingham, AL, Huntsville, AL and Tuscaloosa, AL. 

I’ve had 2 tenants have units in excellent shape for my showing day when they wanted early termination.

Either way, I recommend finding peace with what you can't control (whether that is eviction or them living in the mobile home); otherwise, you let the siblings keep winning if they impact your happiness.

Because they are new to the business, they assume it is a commodity based business and they win business by being the cheapest option.

If I'm in your shoes, this is also an excellent time to document and track each and every transaction + its associated MLS ID.

If both agree, it seems like a win-win as the new tenant gets to move their stuff in over a few weeks and the old tenant doesn't have to pay the full month's rent. 

Good for you for creating a win-win situation! 

Win - win all around.