There are 2200W inverters. Perhaps they have gone 2200 instead of 2000 (2200 is the new 2000? :) ) like the inverter- gen people did.
https://www.giandel.com/products/usa-can-ps-2200kscOP says wire to inverter 120v outlet, but many only have a receptacle. Some have a hard wire option though. If OP goes inverter receptacle to his 50a receptacle he will need a sort of "suicide" male-male job, but I suppose that is safe in this case.
I just brought the 30a shore cord around to the front of the 5er and plugged into the inverter (converter off, fridge on LP) but a 50a shore power cord is unwieldy, so I can see why the OP is doing it that way.
Curious how he gets the 2500w generator 120v output to the 50a receptacle if that is what he does (not clear to me from the OP) Use the same male to male cord he uses for the inverter?