Thank you for your support and information. Last night while in Self Powered Mode the Powerwall put out a peak of 3.7kw. My question for Tesla is why was the Powerwall able to provide 5kw in June when it was just as hot.I would not be giving up on getting this fixed and your posts are the first that I have seen here reporting export limitation in this way. AS @GWord pointed out, the Powerwall datasheet lists 50C (122F) as the maximum operating temperature. Looking at the historical weather for Phoenix, AZ on 8/13/2023 when you first posted the max was 108F which is below the rated specification. Your Powerwall may be outside and in full sun and therefor in a hotter environment, but there isn't any data that confirms this:
There have been issues with Powerwall coolant leaks and maybe that is what is happening, maybe the internal temp sensors are off and that is what is happening, but either of this should be fixed under warranty.
I think that you need to verify that you can get the full 5kW out of the Powerwall in a cool environment. You tried going off grid and turning on the HVAC, but that didn't work as the HVAC startup is too much for the Powerwall to handle. I am very surprised that your AC unit is backed up load with only a single Powerwall. So, instead in the cool morning with your HVAC off, go off grid and just start turning on lights and electric appliances (oven, stove, microwave, etc) to ramp your load from the 0.5kW base up to 3+kW. If the Powerwall can't handle that then there is a serious problem.
Really the worst part is Tesla doesn't provide an escalation path beyond their Tier-2, which are saying the system is working correctly so no further action is possible.
It is curious there aren't really other people with this issue. I wonder if Tesla released a software update to limit output to protect the battery longevity since that is what is under warrantee but then wouldn't there be a huge public outcry?