Is there a label on the generator? A 12V will have a red metal tag on it, while a 6V has a black painted tag. Unless you're running a lot of high current demand accessories like an electric cooling fan, an air conditioner, and/or a high output sound system, a generator will have no trouble keeping up with the demand of keeping the battery charged and the lights working well.
What is the voltage at the battery posts (not the cable connectors) with the engine at fast idle and the high beam headlights on? It should be 13.5 volts or higher, up to about 14.5, depending on the state of charge of the battery.
Jerry
