This time though I will be ahead of the curve in terms of not having to borrow from the bank.  

I don't think the bank would be ok with it, correct?

You started by saying that you have Local bank relationship. 

Community Bank or Big Bank? 

Depending upon whether they are a bank or a trust company, custodians are heavily regulated by the IRS, DOL, possibly the FDIC, State Banking Commissioner, and maybe the Comptroller of the Currency.

Quote from @Mason Jacobs: @Marios Andrianos Im a private lender and would refi you as long as you had a LLC and a few other small things, if you are investing in the United States are you trying to use a bank in Greece for the US property? 

Local Banks & Credit UnionsSmaller banks and credit unions are more flexible than big lenders and might approve you with:✔ A detailed project plan✔ A licensed subcontractor network✔ Proof of rental income history (since you own rentals)4.

Neighborhood Scout is not going to cover any shortfall you have if it doesn't materialize.you could try finding a bank or broker that will do 80% out on a refi, although probably not possible to close by the end of this month. 

For example if the house is worth 400K, but they are willing to sell it for 375k because they are avoiding having to pay 25k in agent fees, then the bank can consider the equity gap as a down payment.

@Eduardo Cambil  Check with local banks or credit unions.