Need battery for house and gen starting?

That's the wrong type of battery to use as a house battery. It'll work ok, but a deep cycle battery will withstand discharges better and apples to apples last longer as a house battery. Idk if you can still get a true deep cycle only 12V battery as most are "Marine/Starting/Deep cycle" but at least they have thicker plates and will last longer than a pure cranking battery like you're looking at.
And the onboard generator will start off of any 12V battery basically. It's just a little lawnmower engine. Any 100ah deep cycle will crank the generator over without issue.