That's correct. I just cut off the lighter connector and crimped on a power pole. When you get your fridge call Truckfridge and buy an extra 12 volt cord. One for the new power pole and one with lighter plug for other uses.
As for in your rig I'd pass on the new lighter socket and just put a power pole on the factory wiring. I'm "assuming" that since the 3 way uses a heater in 12 volt that the wiring and fuse/breaker are more than enough for the Truckfridge. I think the worst draw you would ever possibly see is a little over 5 amps in the max setting from the truck fridge. I'm betting the 3 way draws much more in 12 volt mode. But don't know for sure.
Someone correct me if I'm wrong.
Realistically you could probably just cut the end off the the Truck Fridge power cord and splice it to the factory wiring. Solder and heat shrink or good crimp connectors. You don't really need the extra connector as the power cords just plug into the fridge like a power cord for your PC.
I don't know where you camp. But if you frequent campgrounds with 110 AC shore power it makes sense to run the fridge off that when parked. But you have to access the rear (hinge side) of the truck fridge because the 12 volt power cord has to be unplugged from the fridge to plug in the AC cord.
Hinge side of lid, Controls on top, power ports on the bottom near the compressor.