In my 2004 American Tradition, the outlet that runs the fridge+icemaker is actually split, with one lug powered thru the inverter and one not. The ice maker (separate plug) uses the inverter outlet while the cooling unit does not. However, the fridge cooling unit can be plugged into the inverter outlet and it works just fine. It's just a heater element, so there are no problems with a modified sine (MSW) inverter either.
The logic for this is simply that propane provides a convenient and inexpensive alternate heat source when no 120vac is available, so why tax the batteries by using inverter power? The Norcold 1200/1210 cooling unit has a hefty 120v power draw, 450 watts for the two heaters that drive the cooling process, and the heaters run pretty much non-stop in warm weather. That's about 40 amps from the batteries if powered via inverter. In comparison, the fridge barely sips LP and will run for days on just a gallon of LP.
The ice maker, on the other hand, needs 120v power to operate at all, so if you want ice when off the power grid, inverter power is the only way to go. The ice maker doesn't use 120v power for cooling - the fridge/freezer does that chore - but it needs 120v power to operate the motor and clock that manages the ice cycle. Technically that could be done with 12v powered equipment, but it is simply not economical to build a 12v icemaker just for the RV industry, so it's a standard residential fridge icemaker designed for 120v power.
So, feel free to plug your Norcold fridge into an inverter outlet if you like. If you don't use the icemaker, simply swap the two plugs. If you do need ice making capability, put a multi-tap on the outlet used for the icemaker and plug both cords into that. Either one works fine.