Then he told him to leave. Understandably, the renter didn’t leave, because doing so meant he’d soon be out $10,000. When he refused to leave before this was resolved, the rep told him he was going to have him arrested.

If they tell you to leave, you refuse, and you stay there for the length of time it takes for them to call the cops and for the cops to arrive, you're committing a crime. You don't need to have a formal written trespass in order to be arrested from a private business for trespassing. It happens all the time, often because someone is under the impression as you said that the "warning" they got verbally from the cops 10 seconds ago wasn't legally sufficient. It is.

In practice, the cops when they arrive will usually start off by asking you to leave and informing you for the bodycam recording that you'll be arrested if you don't. If you then agree to leave, you'll almost always be all good. I think the extra flexibility about it stems from some mixture of courtesy, wanting to have an airtight case for the arrest if you do refuse, and aversion to paperwork. But you can technically be arrested on the spot because you refused to leave the first time the business asked you to, and the pattern of "please leave" / calls cops / cops come / "leave or you'll be arrested" / "no" / arrest is a very standard pattern, nothing to sue over.