Ok so here is what I see - you are in at 153k between both loans. With the cost of the rehab at 60 - 80, that sets you at 213k - 233k in loans. IME unless the house is going to appraise in the 213k - 233k range construction financing is not an option. The lenders prefer a LTV of 80%. 100% LTV loans are out there but the interest rate is much higher. While I certainly understand and agree that the bank has a vested interest in the property a loan that exceeds the value is not in their best interest, and they will have underwriting issues to even attempt it. HELOC is not an option since you already have one and there is no longer equity. There might be loans that exist that are Federal that address this sort of situation - I am not sure I haven't experienced the same scenario. I believe you would be looking at an FHA 203K which will look at the value after improvement but unsure if the value is there.
Your other option since you have stellar credit and high income and the way I would probably go is to use another form of collateral such as retirement, stocks, personal savings etc. You won't be able to write off the interest like you can with your mortgage, but the rates are low and you won't have the added expenditures of closing costs and appraisals etc..
Another option if you have retirement investments in IRA and some 401k is to take out a loan from yourself. You will have to pay a low interest on the loan, and can just pay the interest rate with out repaying the principal.
Certainly worth a trip to your banker as there may be other loan programs which would work. There are so many its impossible to know with out asking.