You turn to private lending sources, but they want three, four or five percent more for a cash-out refinance loan -- even though the entire loan is collateralized.

The only things I can think of are if lender has made additional collateral a requirement (and none exists) OR is requiring a particular endorsement to the loan policy which cannot be issued under the circumstances. 

If you have the assets/ income to prove at a bank then chances are that you should stay there, or use the same assets as collateral for the private lender.It is my understanding that a private lender would be easier to obtain financing but more expensive, so that decision is yours. 

Can you cross collateralize?  

There was another property used as collateral which was refinanced when this sold.

If you had a little more equity in the property you are working on there would be some opportunities for HELOC/Cross Collateralization, but as is it will be tough.I think more realistic would be 85% of purchase on a bridge, but with closing costs it would be closer to ~17%-18% when all is said and done.

Why did you decide to use your other real estate as collateral vs doing a 1031 exchange? 

Ask them to text you a picture of their drivers license as collateral.

What is the area like where the collateral is located? 

Typically there is some collateral but it does not seem like there's much to leverage here so that should be sufficient for peace of mind.