Stalling

I feel your frustration, if you didn't find that harness fault on the ac line, I was hoping that's what it would be. I believe there is a battery temperature sensor on this vehicle, but I can't find any diagrams or information on it. This could be an ECM failure too considering the ECM is directly in control of the charging system. Check one last thing for me. Check the back of this Junction box on the driver's side dash, there are current detecting circuits in these junction boxes that the ECM watches.
These charging systems can be tough with a strange fault like this, The High/Low Speed canbus is used between the ECM and instrument cluster which is how the battery charge lamp is shut off after start up. The ECM has to send a command to the instrument cluster to turn the light on, I agree with you that something is causing the charging fault.
You can check the battery by just putting your multimeter on the battery during cranking and watch for its lowest voltage reading. A shorted cell in the battery might not be showing itself until a certain temperature.
I'm not sure if your scan tool has an oscilloscope on it but you could also watch the two wires on the alternator, the yellow/light blue (Generator Comms) and the yellow (Generator Monitor) and see if any signals are dropping out.
Because this problem happens when it's cold, it won't show itself at AutoZone inside where it's warm.
Those 2 wires are the main control for the alternator. I'm very surprised you're not getting any other codes, have you tried a full system scan after seeing the Generator Fault PID, there might be codes stored in other modules you haven't seen yet.
So, there's still some things to check.
I wouldnt even call this an intermittent fault now because you know its temperature related. You could also scan the vehicle with a thermal camera and look for hot spots on wires or components.
If you had a P0622 setting you would know that communications on one of the alternators wires were being lost. If you don't have that code, that can indicate it's not an alternator fault. But definitely run a full system scan and see if there are any current or history codes.

The other thing I noticed in your live data video is that the scan tool was indicating a Generator Fault, but if you notice the ECM was not seeing it, just above that it was not putting the Generator Fault Lamp on. Thats also very strange, it looks like part of the ECM is seeing a fault, but another part is not reporting the fault. I didn't see if you had actually unplugged the ECM and checked the connectors or not.

I fully agree, you were seeing a Generator fault, I saw it too. I don't think it's fixed. But like you said until it's cold enough to see, we won't know. May want to warn your customer about this, just to cover yourself. I have pictures of the Fault on your scan tool. Changing a part due to a code is the worst thing to do. But that was their call. Hope it goes well. If some other code comes back or the battery light, we'll have to look back into it, But the intermittent cold issue is tough, especially with spring coming, it may not act up until next fall now. Anyway, have a nice day.

You're still getting the Fuel Tank Pressure sensor code? Is it still reading 4.99 volts on the live data stream? I really want to figure this out and at least know what's happening here. Because I know I'm going to come across another 500 with the same type issues. What about the Generator Fault, is the scan tool still picking that up?
It's possible since they replaced the sensor, they erased the memory and reset the system. It will take a number of key cycles for certain codes to set. But I'd really like to know what you're getting for live date. If you don't mind, I really don't think it's completely fixed. There may be more than one issue.

Glad to help as far as we got. I'll be poking around to find out more about these cars for future reference. I'm sure I'll be running into them again. Best to you.