Not sure about your generator, but my Champion 92001i has a lever in the gas cap. You are supposed to open the lever before starting the generator. After the generator is stopped, you are supposed to keep the lever open for at least half an hour to allow any thermal contraction when cooling down to happen without creating a massive underpressure in the fuel tank. Then you close the lever to reduce volatile organic compound emissions when in storage (exception: if you always wait before running out of fuel and store the generator without fuel, it may be better to keep the lever open so that whatever little residual fuel is left there has a chance to evaporate instead of having a chance to gum up).
On the other hand, I have two lawnmowers (a new Stihl and an old Briggs&Stratton) that don't have any lever in the gas cap. I assume the gas cap in these cases is supposed to have some small hole to allow airflow. I can tighten the gas cap fully with both lawnmowers, and they don't get starved for fuel.
I suspect your gas cap has some vent that is blocked now. I haven't carefully observed the Stihl and Briggs&Stratton mower gas caps, but I suspect they must have some vent too that's hard to spot because it may be so small.
If there's no vent and the generator was bought used, it's possible the old owner broke the gas cap and installed an incompatible one. You could try drilling a 1mm hole in there, but if you transport the generator in a car full of fuel and you need to do a panic braking stop, it's possible it falls over and the gas leaks from the 1mm hole.
