I have a copy of the DM 4223 installation/operating instructions. PM me, and I'll email the .pdf.
I don't recall nor do the instructions show any inside thermostat. In the Dometic install photos in the instructions I just see that coil sitting there that you called the AC thermostat.
What I do recall is the factory installation came nowhere close to meeting Dometic specs. And the counter above the fridge got very warm, almost hot, when fridge was running.
So I insulated around the fridge, added some Reflectix in the top of the outside fridge compartment, rearranged the gas exhaust, and anything I could do to improve air flow. All helped, but didn't get me over the hump in 90 deg temps, thanks to those "couldn't pass gas in a hurricane" plastic vents.
What did make the fridge reliable in all modes was installing a 12V computer case fan to shove warm/hot air out the top vent. This was detailed 7-8 years ago in a thread in the A-frame section entitled something like "Liking my A-frame fridge" started by me (pgandw). The same computer fan on the exhaust vent has worked wonders on the larger fridge in the HW A-frames (I used 2 case fans on the larger fridge).
Fred W
now 2019 Flagstaff T21TBHW A-frame
then 2014 Rockwood A122 A-frame