yes but it probably would be cheaper to replace the stove. 

And you assume the rest of the mortgage that you're already guaranteeing.Your idea to refinance immediately after to get favorable financing is perfectly fine. 

My concern is that if i apply for a HELOC to use for the down payment before i apply for a mortgage, A) I wont get approved due to my DTI ratio, and B) I wont get approved without first showing the bank I have the 20% down payment before pulling the HELOC.

Supposedly we can then use that 8k plus what we are paying for rent today to pay a 13k mortgage house.

Can you find something cheaper and run by a competent property management company?

But you have to understand the seller is taking on a significant risk of having someone else run the property and not knowing if they will get the final payout.If you are getting typical rates, why would you not just buy the property outright with 20% down and obtain mortgage financing from a lender?

Mortgage lending is highly competitive, and there are many newbie loan officers that will bend over backwards to get you to pull a credit report and pay for an appraisal.

Although we’re not a local bank or credit union, we are independent mortgage brokers with experience in financing options that consider rental income rather than personal income. 

Other than that not too much you can do, the only alternative post-closing would be to offer cash for keys for someone to move out quicker, but if it's short notice to vacate period like 15-30 days than it's probably cheaper to wait. 

Quote from @Sushil Iyer: We have a SFH and a duplex that we own free and clear, and a duplex on which we have a residential mortgage.