What does the battery accomplish? I mean, maybe an UPS for your computer if you're running something mission critical? Something that can last for 20 minutes while the generator boots up?
But I don't think most people are running mission-critical programs at home.
In my home, a single Powerwall battery could cover 90+ percent (or perhaps 95+ percent?) of nights even in an off-grid arrangement. Covering the rare extremes with a $150 generator would be more cost-efficient than using a larger battery, and likewise, covering the usual case with a battery would be more cost-efficient than running all nights inefficiently using our $5/USgal gasoline.
But I don't think most people are running mission-critical programs at home.