You are right, no thermostatic control on 12 volts. The wiring of that
internal fridge switch is not really covered in any detail in the
Bentley camper diagrams but there should be a sticker on the top of the
fridge that shows what is what for the fridge wiring. You should be able
to test the 12 volt element for ohms and tell if it is open. Also if you
measure the 12 volt wire's voltage at the switch with the switch on 110
and then switch it to 12 volts you should see a drop unless there is a
problem beyond the switch.

What kind of readings should I expect to see in terms of Ohms? I guess I could compare things to the 110 side, huh?  Does infinity means a broken wire, or do I have that backwards?  I should know this, but it's been years. I do have a small wiring diagram from the fridge manual (small being the keyword) so it's hard to see what's what. I do remember the wiring diagram on top of the fridge though. I actually just had this thing out of the van about two weeks ago for a cleaning and fan upgrade. I'm becoming an old hand at getting this fridge out of the van! I think I've got it down to about 15 minutes;)

I have seen several Westy fridges with cooked 12 volt wiring due to the
fact that they can get left running on the 12 volt setting for years
while the vehicle is driven daily, unknown to the owner that this is a
waste.

Mark