Your best bet is some kind of bridge loan to close efficiently and then refinancing once it has been seasoned to take full advantage of that discount that you are getting. 

Financed with an FHA loan and mortgage being right about $2,000 monthly, included escrow for insurance and homeowners' insurance.

That means you will not be able to sell it to someone using a FHA, Conventional loan etc. 

Since I am now the "bank" or "lender" I have all the rights of that lender to enforce recovery up to the full outstanding balance of the loan.

Even if you do a cash out refi, you still need to pay off the initial loan, and then will not have enough money to pay all cash on the next and so on when prices are at a high premium.   

The state paid about $35,000 down for us and our mortgage was about $820/month ($123k loan amortized over 30 years at 4.5%).

Also, do the math - does demoing your current house + paying off the current mortgage immediately, then getting a construction loan make sense?

If you can find a vacant lot the bank includes the land cost into the loan so you save money in the long run. 

Hi Nolan - Have you explored DSCR loans?

@Nancy MillsMake sure you have the person also sign a personal guarantee on the loan --- that's pretty normal.