Hello Souter, Congratulations on your new to you Redwood and welcome to the forum. We also have a 2012 36RL but have a Norcold refer in ours. Yours may have been changed out. I did find the manual for your Dometic 1350 here: https://bryantrv.com/docs2/docs/Serv...,%20RM1350.pdf
I didn't see the E4 error code in the diagnostic section. Most 12v appliances are picky about their power. I would check under the slide out room for broken or damaged wiring. I would also remove all the 12v switches carefully to look for damages there also. The wiring is inside the aluminum tubes and can get chafed damaging the insulation possibly causing the shorting out of the wiring. I know that the 110 volt wiring has a junction box located behind the left bottom drawer left of the fireplace. To remove the drawer pull the drawer most of the way out and locate the black tabs on the drawer guides, push one up and the other one down and the drawer can be removed. Not sure if 12 volt enters that same end of the slide out or not. You could also check behind the bottom drawer under the refer to see if the 12 volt connections are there instead.
Let us know what you find out.
Chris & Maria 2012 36RL