After encountering identical E1 error codes on my Dometic Fridge (Model RM 3762), I contacted my RV dealer who recognized the issue right away. In my case the E1 error code would appear in both LP and AC modes. The fridge would in fact work normally for about 6 hours or until it reached the set operating temperature and shut off as normal. The problem was that it would not re-start and I would constantly have to manually shut it off and re-start it. This procedure had to be repeated over and over every time the fridge reached operating temp.
My dealer had stated that the issue was with the temperature probe or better known as the "Thermistor". To confirm this, he had me push the wire of the thermister through to the back of the fridge and leave it exposed to the outside air temperature. In doing so, the fridge worked normally but would not shut off as expected and would natually freeze everything over time. However, this confirmed that the E1 error code was an thermister (temp probe) issue. The thermister is located inside the fridge and is attached to the "fins" and the back wall of the fridge. It is wired to the electronic control panel in the back of the fridge. I have replaced the thermister (under warrenty) and have had no issues since. (the thermister I removed was subsequently tested at the dealer and confirmed to be faulty. Hope this helps everyone who I see has had this issue.