You'll also need to show stable income for the past two years through W-2s or tax returns.Talk to 3-5 lenders to compare their rates, fees, and terms.

I know of several DSCR and portfolio lenders who offer different terms based upon your situation.

I also require additional security deposit and sometimes a guarantor if they were approved in this manner.I also agree that this situation sounds like a pattern not a one-time mistake or short term tough situation that caused this.  

I was last getting terms around 5-6% for the first 5 years, then floating with prime, 20 year term. 

I'm born and raised in Houston and now work with a full service long-term rental management managing over 300 doors - all SF or MU (up to 4 unit).

Part of the motivation for congress writing Dodd Frank was to clamp down on the investors offering preditory terms re  rent to own and seller financing and that view has carried into craigslist rent to own adds and you say zillow. 

For long-term growth, consider newer single-family homes in high-appreciation suburbs or well-located small multi-family properties.

Specifically, short-term rentals can help you get REPS status.

They are sure to move on at the end of their lease term. 

My main concern is that you underestimate how much time/effort this will take.