The decision to use a GFCI breaker was a correct one.
The problem is this style of transfer switch is about as "Bangkok wiring" as you can get - the worst series of code violations I have ever seen in a consumer product*. In modern panels with modern needs, they're a very bad fit, and this is a case in point.
These days, hot and neutral need to be handled together on a per-circuit basis. This thing doesn't do that, and that's the problem.
A better transfer switch (cheaper, too)
Obtain another service panel. Get enough spaces for every circuit you might conceivably ever want on generator or PowerWall + 4 more spaces. Spaces are good. 40 spaces is not excessive. Prefer the brand and type that matches the GFCI breakers you already have, because then you won't have to replace them (ouch)!
Then, add a generator interlock to it. This is a moving piece of metal that prevents two double breakers from being on at the same time - a regular double breaker, interlocked with either the main breaker or another double breaker.
Siemens makes the cheapest generator interlocks, Square D "QO type" is the runner-up, both for main-lug (no main breaker) panels. Code requires the interlocked breakers be bolted down (Siemens' kit does this). Some jurisdictions require it work with the panel cover missing (which eliminates some types).
Shop carefully for your panel and interlock, balancing cost of interlock vs cost of not being able to reuse breakers. Look at the breakers, not the panel: if you have alien HOM breakers in a GE panel (bad), look at HOM panels not GE.
Installing
Link it up with the original panel through several rigid metal conduit pipes. If they're within 24" you only need 1-2 large ones.
Install the generator interlock. If this panel has a main breaker, reserve that for utility. Make the interlocked breakers large enough to supply the subpanel, but any size you please - they are just switches and their breaker ability won't be used. 60A breakers are the cheapest.
Come off your main panel with thru-lugs, subfeed lugs, or an actual breaker. Use wire big enough to be safe at the max possible amperage. For instance if you lug off a 200A main panel to a 200A sub, your max current is 200A so the wire must be 3/0 copper or 4/0 aluminum. It's fine to have a 200A, 3/0 wire feeding a 150A subpanel breaker, as long as the wire physically fits; you can't whittle it down.
Move branch circuits (hot(s) AND neutral) over to the new panel, either by moving their supply Romex to the new panel, or extending hot+neutral into the new panel via the conduit pipes.
I'm not really addressing removing the neutral-ground bond on your generator. This type of installation renders that a non-issue operationally but it's still a Code issue.
Operation
To switch, shut off all your breakers. You generally want to switch no-load or low-load. Then shut off the present source, then switch on the desired source. Then turn on breakers of the things you want to power.
Why does this play nice with GFCI? Because you're making the generator selection before the GFCI breaker. In both modes, the GFCI breaker pulls hot off the bus and neutral off the pigtail, and supplies protected hot+neutral to the whole circuit.
Footnotes
* You couldn't hand-build a transfer switch like this, you'd get a sea of red-flags. I suspect it's in a gray area between what the UL White Book allows in an appliance and what NEC allows in field wiring.
The general concept of these transfer switches is there is a row of SPDT switches. One side taps the generator hot, the other side goes to the panel and taps a circuit's hot off the breaker, and the "common" goes back to the panel and joins the hot wire that used to go to the breaker. There is one single neutral wire that handles all generator current. That's ... ghetto, but workable for a 1960s panel. Here's what it looks like inside.
Look, ma, no neutrals!
