By its nature, it also requires trespass, but not 'criminal' trespass as another poster erroneously suggested.What you have is likely a stubborn trespasser.

I feel like too many people here are old school and are too stubborn to ask for help, or they are so deep routed in their father's friend or their uncle who got burned in real estate that no one looks into it.

Next time do not remove that inspection contingency or work with an agent that guides you differently.The loan contingency is what you need to lean on, get with your lender and let the agent know that you want to cancel due to something along the lending side so that you have backing if the seller tries to be stubborn about the whole thing.

It kinda makes since but to me but it also seems like I'd be rewarding somebody for non-communication and stubborn.

In Houston we try to offer cash for keys a lot but some people are just stubborn and want to punish you by making you go to every effort to get them out.

It can be done without the use of a lawyer although with fairly stubborn tenants the use of an attorney can sometimes light a large fire under the real ends of stubborn tenants.

SOMEONE THAT ALLOWS AN EVICTION PROCEEDING TO ESCALATE TO A JUDGE GRANTING A POSSESSION ORDER IS STUBBORN AS hell!

(I used to get peeved at threat of suit (it still can bother the wife) that I would take a stubborn stance (and still never got sued) but now I typically ignore the threat (except for one recent threat that concerned my primary residence where I consulted a lawyer - no suit yet).Regardless of your intent of the pet deposit, I would use it as a pet deposit to pay for any pet damage and refund the rest.

I don't know why he's so stubborn about this.  

You can't reinvent the wheel basically.Lets be honest, FSBO are usually stubborn thinking they have a property worth more than what it really is worth.