With a full doc loan, they would like to see a signed lease, with a stated it may not be an issue.

This reminds me of a problem I had about 5 years ago when purchasing a property for rental. It was previously owner-occupied, so there was no current tenant. About a week or so before closing, the lender requests a signed lease for the property to prove income (full-doc loan). My PM company will not advertise a property until I own it. In fact, I am not even sure I can legally lease a property to someone prior to actually owning the property, without adding some additional legaleze to the lease agreement.
This can create a catch-22. At that time my solution was to find a friend who was “thinking” about moving out of his home to remodel. He signed a Month-to-month lease with me that didn’t start until about two months after we planned to close. The lease also allowed that he could get out at any time with 30-days notice. It turned out that after we closed, he decided to wait on his remodel and canceled his lease agreement with me.