This would be my first commercial property and while it has some tenants already I know these can be a bit more difficult to lease out than a 2 bed 1 bath. 

Rehab and get the house rented.You can even start the refi a little before you finish the rehab if you want - assuming you have a renter lined up and ready to go.I once had a refi done on a house I bought within 45 days or so of the purchase.

No matter what the situation is you have to make it to the finish line... 

I have 250K on a line of credit with interest at an Annual Percentage rate of 5% that I can utilize.

I hand it to all you guys that are able to invest across 6 state lines. 

The offer is accepted and we have about 30 days to close.It's a 3 Bed, 2-1/2 bath single family home that is expected to rent for $1,500 in our market.

Take out a 100% Home Equity Loan (not line of credit) from my credit union on my personal home and put some cash with the equity loan for a cash purchase.

Other contractors I've spoken with are lined up 3 months out.

I’ve always heard the refinance discussed as a mortgage - but I’m wondering why not a Home Equity line of credit instead?

All of them have  Residential loan products for investment properties.  15, 20, and 30 year mortgages.You can also try on-line Lenders like LendingTree or Quicken Loans.