If I'm understanding your question correctly, you wish to use the inverter to start and stop the generator. If that's correct, and the chargeverter is plugged into the generator, then the generator will just start when the dry contact on the inverter switches on, and then off when the inverter sees sufficient SOC. The CV doesn't need to be connected to the inverter at all - just to the batteries, and then only through its output cables.
You probably should program the CV to stop charging at a certain battery voltage, only as a safety, in case the generator never shuts off, based on the battery voltage it sees. Voltage is a poor way to monitor a LiFePo4 battery but OK as a safety against overcharging, in this application. Otherwise, the CV will just turn on when the generator turns on, and off when the generator turns off (because it has no power). The inverter is controlling generator start/stop based on the battery SOC, and the CV is just going along for the ride.