Ok so I solved my problem, at least temporarily. I turned off the power to the heater and with a sharp utility knife I cut the membrane just below and on the sides of the buttons. I then carefully lifted the membrane just enough so I could expose the contacts of each button to fit the spray tube of electro-contact cleaner and spray some under each contact. Then slid a flat piece of paper towel under the membrane of each button to remove excess cleaner.

Seems to have worked like a charm and we'll see how the repair holds up.

I may tape the membrane to protect it or figure out some other way to seal off the buttons. I just don't want to seal it so tightly that when I have to remove the tape it pulls of the buttons completely.