Grid tie inverter on Generator Set
The power to run your charger is coming from the inverter, not the generator. This is because the inverter produces a voltage a bit higher than the grid voltage (the generator in your case). This is so the inverter can provide power to the grid when the local load is sufficiently low. I can't see a way to force the charger to take its power only from the generator with your setup. My suggestion is: don't connect both the inverter and the charger simultaneously. Connect only the charger when the load is low and the battery needs charging. Connect only the inverter when you need to power a high load.