One thing to look at is the settings in the inverter/charger is the input frequency. MagnaSine, and Schneider and many others do have an adjustable input frequency range that the charger will tolerate. My MagnaSine does have a very wide range and can be set wider than the factory setting of 58-64 hz. Set the range wider...say 52-68 hz. This will allow the charger to allow for “generator stumble” where the generator will drop in speed/frequency when a load is applied.
A second option is to use an auxillary charger such as an Iota which is “switch mode”. These do not care about frequency as much as a solar inverter/charger. The bad of this approach is that these (Iota type chargers) are not “power factor corrected”, requiring a larger generator to provide the charging power.
Im using a Onan RV generator which has a far larger rotating mass than most lightweight generators as a result has no problem with sudden loads applied
The MagnaSine does have a charge ramp up time on charge, unlike some units which go from zero to max in a few seconds. This allows the generator to adjust for the loading. All the Schneider units that I have seen also have this feature.