The PowerWall is inherently not good at working with a generator. What you really need is a separate AC input to the system for the generator that is free running and not synchronized to the battery inverter output. This is what Hybrid inverters do - they basically have a separate battery charger running off the generator input. I believe someone said that Tesla is doing a trial with a handful of customers that have generators to see if their preliminary solution works as expected. I assume they are nesting the Tesla gateway inside the generator's transfer switch so that the Powerwall can synchronize to either the grid or the generator. The only question is how they are controlling when the generator starts. Most would automatically start after a few seconds of grid outage. You wouldn't want that to happen with a PowerWall.
Indeed. When they had a DC only version, it lent it self to integration with the hybrid inverters (not like there are a lot of choices there). But AC Powerwalls don't work with those and require a complex control system to avoid nasty things from happening. I am in the process of building a house, so if Tesla disclosed this approach, I could at least design that in to the electrical system, so I could add powerwalls when the software and hardware were ready.
But as of now, I can't do anything, and because of the closed architecture of their system, other companies that do inverters can't really incorporate them into designs either. I am puzzled by what they are trying to optimize for.