January rent has not been paid to the property owner, as I am sure the lease spells out, but to someone else.

Or just require a full year lease to students?

It's difficult to see how a tenant will be after screened and signing a lease.

I advertise via Sabbatical Homes and Harvard Off-Campus Housing and insist my leases begin on Sep 1, Jan 1 or Jun 1 and end on Dec 31, May 31 or Aug 31. 

This raises a question because we had a lease agreement on file showing a rental amount of $2,000 per month."

Master Lease with Option. 

If you can rent it out, get it rented out, get a year lease, then start approaching commercial lenders for either a DSCR loan or a 2nd mortgage, with cash out. 

Here are a couple of ways you could structure the deal:Lease Option with Seller Financing: You could lease the property with an option to buy later, allowing you to rehab it while giving the sellers time to find a 1031 replacement property.Seller Financing with Deferred Payment: Structure the seller financing with no immediate payments, allowing you to refinance once the property is appraised, and the sellers can complete the 1031 exchange when ready.Escrow Holdback: Transfer the deed into an escrow account, giving you control without immediate transfer, so the sellers can take their time with the 1031.Third-Party Facilitation: Use a qualified intermediary to hold funds and facilitate the 1031 exchange while you refinance.These options allow flexibility for both you and the sellers.

If you want the furniture, just negotiate that and put it in the contract.Could you lease it from them for those 3 months and operate under their permit till yours is available?

I have REPS status, but currently rent this particular Property to my Property Management LLC that leases this home "from" me and and pays me 5000/month rent as shown on my soon to be Schedule E.