You are right 5% is very low, that's why we use this strategy (minimal cash out of pocket with the best available rate), you are also right that we have to pay PMI, but the PMI is minimal on a conventional loan if you have great credit, it's only about 0.2% of the loan amount per year, so I pay $72 per month on this one.

Try local banks and Credit Unions, they'll offer the best terms for you. 

What documents do they require, what credit scores do they allow, how do they verify previous rental history, etc.?

@Kent KettellWe have been using loan to cost funding our deals the past three years, with community and credit unions, and the refinancing once stable to agency.we are able to get 80% of the rehab built into the loan, and it has allowed us to come up with less.

Ideally this happens without any seasoning/payment history period, since I am reselling the house and lose title upon sale.

The only things I change are that I require a copy of their housing voucher (which tells me which agency they're with, how many bedrooms they're qualified for and if their voucher is still active) and I require only 1 months rent in income instead of 3.The rest is the same - I call every landlord they've had for the last 5 years, I check their background, credit, etc. 

Things like income documentation, background checks, credit reports, etc mean a lot more to me.

This was not on credit history but on chain of title. 

If your tenant has a history of failed payments, we permanently move to 3 day holds to ensure the funds are available.

Your offer should have included a requirement that the Seller transfer all tenant documents (application, payment history, lease, addenda, etc.), spare keys/openers/codes, prorated rents based on the day of closing, security deposits, and anything else pertaining to the rentals and tenants.