When I did the swap on my vintage rig, I ended up removing the full front bezel to get the fridge out of the older unit (This required cutting alot of glue to separate the face plate from the thermoformed plastic interior casing), so I could side the unit out without having to dismantle the cooling unit, then do the reverse to put it back in on the other camper. (I swapped the fridges between my 1974 KIT and 1975 Amerigo, as the KIT's fridge was less rusty).
Even with that bezel removed, I had to complete remove the door and its jam assembly, plus the bathroom door in order to squeak it out.