Note: i'm not a generator expert or an electrician. These are my thoughts and mine only. Do what you will at your own risk. No Holiday Inns were used.
You could adjust your generator a bit, so it's making more reliable power at the top end of loading.
When I recently went through and did a Big Service on my generator (because it asked for it by running like crap last time) I also took the opportunity to dial in the voltage output while under load.
Before adjusting I was seeing 122V @ no load, 115v @ 2200W standard load, and 112V at near full load.
After adjusting it puts out 130V @ no load, 123V @ 2200W standard load, and 118V at a 6000W test load (4x 1500W electric space heaters on full.)
(note I'm pulling these
#s from memory, and might be a bit off. I will try and find some notes I might have written and update this post with corrected data if I find it.)
It no longer sounds like it's working hard even at full load.
My suggestion would be, figure out what it's running at with your standard load, and try to dial it in to run about 120V on each phase, and maybe a touch more. In my case it was just turning the governor screw *just* a touch to get it to bump up a couple volts.
120V "standard" allows a 5% swing, so 126V-114V is considered "acceptable" from your primary energy supplier.