ICE turns together with AC compressor

It's an indication that your hybrid battery is getting weak. If you haven't already begun, start making the preparations for a replacement.

The AC compressor takes a lot of power. It draws it from a combination of the hybrid battery and the motor-generators in the hybrid system. If the battery hasn't got much to offer, then the balance must be made up by the engine.

Once the cabin is close to the target temperature, the load is decreased and the behavior will seem closer to normal.

Take it as a warning shot.

 

They've provided something like that to the on-board battery ECU, so that it can make its own go/no-go decision about battery health.

I understand (from reading) that one of the key data points is a 0.3V difference from one module to the next.

I'm not aware of any place where Toyota has published this stuff in plaintext so that you can bird-dog the battery ECU, but it could be out there somewhere.

The way they designed it, the system will try to mask early symptoms (by tagging in the engine earlier & longer) to let you extract the maximum lifetime from the traction battery. Then one day it will decide it has finally had enough, lights up the "check hybrid system" warning and leaves you a P0A80 code to read back later. That's your cue to $pend the big dollar$.