It seems to me that you could quitclaim
title to the property to your father, which would make him sole owner. Then, you would probably have to wait at least six months, because lenders need to have title "seasoned" over time in order to ensure that they are not being tricked into doing a transaction that they would otherwise not be able to make.
Then, your father would refinance the property in his own name, and pay off your old loan. I don't see any reason why a lender would not do this refi, based on what I've described. However, there is no way to accomplish the same thing over a matter of a week or month, because of the "seasoning" requirements. And, if your dad purchases the property and pays the 20% down, then there will be all sorts of closing fees that will increase the transaction costs, and that will nullify the improvement in interest rates.
I realize that you may be thinking that six months from now, interest rates will have soared to new highs. Regardless, I don't believe that there is any option, other than to take that risk, if the goal is to maximize the transaction savings.
Hope this helps.