In this case would I qualify for about a $4600 mortgage using 75% of the rent income and 40% of my own income?

I have helped clients in your exact situation in the last few weeks.You do not have to refinance that loan to purchase another home.In fact, you can purchase a new single-family home with 5% down and turn your current home into a rental.You can even use the rental income you will get from your current home to help you qualify on the purchase of the new one!

Do you show enough income on your 1099's or tax returns to cover a new mortgage payment. 

Also, focus on 2 years of job/income stability.Class D Properties:Cashflow vs Appreciation: Typically, all cashflow with zero or negative relative rent & value appreciationVacancy Est: 20%+ should be used to cover nonpayment, evictions & damages.Tenant Pool: majority will have FICO scores under 560, little to no good tradelines, lots of collections & chargeoffs, recent evictions.

Since you've got a great source of income outside of that, you stand a chance, but the rates and terms you'll get from a bank or credit union for a non-owner-occupied investment property won't be as low as you think.

And then it will cost you a lot of time and effort to compensate the loss of income.

Typically, if we are in a situation where the loan amount exceeds 75% of the ARV, the borrower has to bring additional money into the deal. 

The income from the property more than covers the bank financed mortgage and the repayment of the down payment to the seller which I proposed to make monthly payments on. 

I am moving out of the house in June and intend to rent the other two bedrooms to two additional tenants, unrelated to the first tenant.

I think it depends on goals and experience, section 8 is a good guaranteed income, air bnb or short term will have higher rents but you will have more monthly expenses (cleaning, internet, lawncare) student housing is another option, and then good old LTR.