I have a Honda 5K that I use to run the well pump, the fridge and freezer in my house during power outages. It can run just about everything one at a time save for the furnace.
I don't take a generator camping. Nor am I a generator mechanic so I'm not going to be much help, but...how old is the fuel in the tank? How often do you run it solely for preventative maintenance? It sounds to me..and again, I"m not a generator wrench, it sounds to me as if something is plugging up the hoses.
I run mine quarterly with a load on it..usually something like a Shop Vac or if nothing else a 100w rough service trouble light. I let it run for about twenty minutes, unplug the loads and then, I TURN OFF THE GAS and let the generator run until it runs out of fuel and stops. Only then do I shut off the on/off switch and THat one thing, I think, has really helped keep the lines cleaned out.
Given the work, the time, the parts you've thrown at it and the aggravation you've suffered with the 2K, I'd bite the bullet and take it into a shop and have it looked at. Maybe it's a lemon, although Honda generators have a good reputation.
These are the voyages of the small ship, "Grus Egg".
2011 17" Spirit Deluxe, "Grus Egg"
"Civilization began when we stopped eating horses and began riding them."