I, too, disconnected the cable from the touchpad and something happened as it now stuck on propane and won't switch to either battery or AC.  For right now, this is OK as I need it to work on propane for me.  But this is clearly a problem with the circuitry in the touchpad  switches, and not with the refrigerator actually working on any of these energy sources.  I'm afraid to disconnect the touchpad again because if I need it stick on anything, it is the propane.  Thanks everyone for your suggestions.  Still trying to figure out how to get a new board for less that the $300-400 everyone is quoting.

Try removing the connector from the flat cable to the touch panel, cleaning the flat cable with rubbing alcohol and a little agressive rubbing with a q tip, then dry and reassemble. Seems to help for awhile most of the time. 

I had just the opposite,  the RM8501 worked on AC but not on propane or battery,  mine is from a 2017 rpod.  I called Dometic and they wanted $427 for the control board!  so   I removed the board from the control panel,  disconnected the ribbon cable and other wire plug and removed the pressure sensitive touch panel,  reassembled it all and it now works ok for AC, Propane and Battery.  Hopefully it was a one off situation and reseating the board resolved.  We found this did not work after we returned from a trip and planned another adventure,  worked ok on the trip but went to prepare for another trip and found the problem.  go figure.  Hope this helps

john

I have the identical problem in my 2016 179 with the RM8501 Dometic fridge.  Tried everything.  

The fridge is cooling on battery, just as yours.   Just won't switch to AC or propane.

Did you ever resolve?

We also had a similar issue with the eyebrow control board several years ago-we ended up replacing the entire unit.  It was about $200 more for the unit installed with a warranty than to buy an aftermarket eyebrow control board and installation. As I recall, the entire unit installed was about $1100 and the control board was about $850

We have had no further issues.

Vann

If you do a google search for rm8501 manual one of the links is the manual in pdf format. There is a basic troubleshooting section but also the error codes, which might be helpful. 

Have you checked the AC outlet behind the refrigerator to make sure it has AC power? If it doesn't, is the GFI outlet tripped? Is the breaker for the circuit tripped? Is the trailer plugged in?

Were you able to switch it before? Are you pressing the button once? twice? It may take two pushes of the button for it to switch. The fact that it does work at all is good. The default is to return to the last operating state when starting up. Beyond trying these things, you can look at this troubleshooting chart. It isn't specifically for the RM8501, but it does explain how the refrigerator works quite well.