I can't say for the Dometic. I really didn't investigate their 3 way as they were so awful to deal with that I decided to buy a Norcold. Sometimes I shoot myself in the foot. I got a Norcold 3 way on Amazon for a very good price. When I went to install it, I found that it had a quick plug that had both AC and DC in it, but there was no clear wiring scheme. I went to several Carson City RV shops and no one recognized it. Finally, DJ's RV in Carson City installed it but without the DC cooling circuit connected. (The fridge needs a low amperage DC to run on gas and that's on a different circuit.) I figured out what the problem was when I spoke to a guy who is the go to person for this sort of thing in the Reno area on the telephone. The fridge is designed for installation in a full RV like a class C. I got the wiring diagram to connect it to the TV, but it just wasn't worth the hassle. It works fine on gas.
If you would like the TV connection wiring diagram for the Norcold, I can send it to you via email. If the installer didn't say anything about the DC, you may want to ask about it. Also, you are welcome to pm me and we can arrange a telephone call and I can explain this a little better. It's a little complicated. Also, you're welcome to come and see the installation since we're practically neighbors.