Welcome to the forum. Tough question to start out with!
1. The 2400 did not come with an inverter. There would only be one on the boat if the previouys owner installed one.
2. As to the question of recharging the house batteries. That's a difficult one to answer. There are many variables to consider, e.g., what appliances and electronics will you be running off the inverter. Things like a coffee pot or toaster oven will pull very large wattage. A microwave over typically pulls 750 to 1000 watts. The inverter would power them, but the batteries would be drained pretty quickly. Rather than run off the inverter, I would run directly off the generator. The house batteries, if in good condition, will keep the lights and and refrigerator running all night. Just be sure to position the battery selector switch to pull power just from the house batteries, so you'll have starting power the next morning. I'd also consider a slightly larger generator. 1.5 kw is very minimal to run AC appliances (except maybe a TV/DVD player).
3. If you put a non-marine generator on, be very cautious of CO fumes (well be cautious even with a marine generator). But non-marine generators don't have the protected circuitry to prevent random sparks, etc that can mix with gas fumes and have disasterous consequences.