Long term buy & hold rental investor. I've hit a snag and need refinance information and leads.

Looking to refinance a rental portfolio in Gaston County, NC (Charlotte Metro). I have called about a dozen lenders and only 1 offers refinance on existing rentals. However, that lender will only loan on LT-Purchase price (not value). So I purchased properties around 70% below market value but they require I bring a 20% down payment on the purchase price, not the market value. (kind of feels like I'm getting punished for buying well, but oh well).

Greatly appreciate leads for long term 15-30 year amortization loans that will loan on the appraised value. I'm fine if it is an ARM. I'm being told over and over by big and small banks "We don't do those types of loans anymore, period."

Details

1) 1 SFR for cash out refinance

Purchase price of home: $35,000

Rehab loan wrapped in

Total payoff amount: $43,600

Appraised value: (don't have official appraisal just BPO): This house will appraise between around the $75,000 mark. 

rent: $935/month

2) 7 properties held in an LLC for long term refinance

Avg. purchase price: $36,285

Avg. appraised value: $50,285

Total purchase price: $253, 995

Total appraised value: $351,995

rents: currently $3,850. Long term should settle around $5,750

My primary concern in both scenarios is to minimize cash to the closing table. I don't want to buy at a 30% discount and then put 20% down on my discounted buy.