So this is somewhat speculative, but just hear me out.
When you're pulling lots of power off your inverter while the truck is idling, your amperage draw will exceed the capacity of the alternator at idle. But that's okay, because the over-drawing causes voltage drop, and the ECM reacts by increasing the idle speed to get the voltage back up.
But, if the hot side of the alternator powering the inverter is connected
only to the inverter (i.e., it's isolated from the rest of the truck, including the ECM), then the ECM will not be able to detect that the voltage is dropping and hence won't increase the RPMs to compensate. This will likely result in the inverter's low-voltage cutoff being exceeded, and the inverter will shut down.
Again, that's rather speculative, but the fact that all my stuff is bonded together and works perfectly leads me to advise that you do it the same way.
As for the wire size to the battery, I would probably just use the same gauge that is used on the OEM alternator, which I suspect is 2 AWG. But I would use 4/0 for the alternator-alternator bond, if you choose to do one like I did. I recommend doing this because if you don't, then some of the current to the inverter will be pulled from the other alternator, and you don't want big amperage going through those 2 AWG battery cables.
Also, you ran your ground cable to the second alternator. Any issues with running it from the grounding lug on the engine block instead? The passenger side battery is ground to the engine block near the bottom of the engine and there is a straight shot from the bolt to the frame rail running to the back seat, so it would save about 4 feet of cable.
I went straight to the alternator housing because I wanted as few cable/lug interfaces (including the physical connection between the alternator and the block and any intermediate brackets) in the circuit. I think going to the block is probably fine though.