What model is the Dometic fridge? Do you know what the DC current draw is when on DC? Most RV fridges don't operate on DC (except to power the control board when on propane)
DC absorption fridges draw a LOT of amps. That could by your killer. Also, often automotive alternators don't provide a lot of charging current to deep cycle batteries.
11.99 volts is awfully close to dead. Try not to let the no-load voltage drop below 12.2 Volts.
I just Googled Dometic AC/DC refrigerators and found that current Portable fridges draw from 2.75 to 4.5 amps depending on the model.. Running continuously for 10 hours would take 27.5 to 45 Amp Hours from your battery.
I assume that you have one of the portables.
I forgot to ask what the coach battery is. I am guessing a group 27 Marine/Deep Cycle which will give you roughly 100 AH to completely dead, so 50 AH or so to 50% (12.2V)
Also remember you have to put a few more AH into the battery than you took out.
You MAY have a battery problem, you MAY have a charging problem or you MAY have a usage problem, or a combination any two or all three. A little detective work should sort it out.