When your generator is running, can you run the A/C and other 110v appliances without any issues?
If so, as others have said it is likely not your generator but I can see why you are looking in that direction.
When you engine is running, it uses the alternator to charge the house batteries so you are not using your Converter / Charger. When you are on SP, you are now using your Converter / Charger to charge the house batteries.
When the generator is running, it is using the same Converter / Charger circuit to charge the house batteries. If you are truly seeing over 13V when charging with the engine and SP and your battery voltage does not fall off a cliff but you are not getting over 13V with the generator, there could be an issue. But I would expect you would see other signs of voltage issues when running the generator.
I would do what was previously recommended and first test the batteries to verify they are not an issue. If they are several years old and have not been well maintained (by you and by the charger), then they could be a big part of the problem.