After a month of dealing with this, countless hours spent on the phone with Apple, and then the senior support care person ghosting me I finally got a decent Senior Tech who read through all the notes and basically said, "enough with this garbage. You've done everything we've asked and have provided more than ample enough proof there is a definitive issue with your phone." He then did something with my account and case that forced Apple to either do a repair/replace on the phone even though it's two years out of AppleCare.
Went to the local Apple store yesterday, and after another 90 minutes walked out with a new phone (same exact model, 6s 128GB) that was loaded with iOS 12.4 that they upgraded to 13.1.3 even though we know it will have issues.
I can't say I'm really seeing any improvement even with a new phone and battery at 100%. I'll give it a day or two to balance out after the upgrade and restoring all my data. But I don't have any confidence the problem will be resolved.
I also still have the issue where the Apple battery case does not work correctly with the phone. While in the store I showed it worked correctly on 12.4 and then when the phone was on 13.1.3 it was dorked up. There is no way they can refute that iOS 13 has a lot of bugs still.
With the case if you place the phone in it, it will not recognize the battery to start charging the phone. If you bounce (power cycle) the phone while it is in the case it will recognize the case and start charging once it has booted up. But if you take the phone out for even a second, that all goes out the window. The other way to make it work is to take the phone out, plug in the case to charge it, wait for the LED to come on (either green or amber), then slide the phone back in. Then it will recognize the battery case and start charging.
Now, on my backup 6s that is also on 13.1.3 it does the exact same thing with the battery case, so I know it is the OS and not the case. Just for testing I upgraded the backup phone to 13.2 beta. Now the battery case seems to work as it is supposed to. But I'm not convinced that specific problem is resolved. Keeping that in mind I can't condone anyone updating their iOS to a beta version. It is bad enough we are all beta testers even with Apple's "released" OS's. Going to a beta version would increase the risk of more issues manifesting. If I didn't have a backup/testing phone I would not have pursed that course of action.