There are a lot of details you haven't filled in in this post (which is fine), but you will likely need a key principal to co-sign loan and a contract with an experienced property management company.The key principle will co-sign the loan to help show the net worth and liquidity needed for the underwriting department.  

I have a cousin in Florida that is moving and would love to come to an agreed price on their place but just take over the mortgage instead of securing my own loan as I am looking for it as an investment and would be required to do 20% down (which I would not have). 

Assuming you can cover a decent portion of the $60K from the 401K loan, get some sort of loan for the rest.  

Further, how do I demonstrate proof of funds on a HELOC (ie, the equivalent of a pre-approval letter for a loan)?

Conventional loan How did you add value to the deal?

He couldn't sell the property because no bank would give a loan on it with that in the contract.2.

I was one of the four original investor to raise capitol to get the loan for construction.

Once we fix our side up and move in the property should be worth $200k-$250k (it was very undervalued at time of purchase).Why not take a larger loan initial on the purchase of the first duplex and lower your down payment, use the cash you have for the second purchase.  

I have 20k saved and plan to use an FHA 203K loan and house hack to net zero/cash flow minimally on my first investment.

Current ideas are to view it as a partnership, or an interest-bearing loan from me to him.