So I rebuilt a wrecked BRZ last year I bought at auction and it was missing a key, also a key fob with push button start. Different company and system, but I ended up learning a bit about dealing with replacing modules, and fighting with everything for months to get it back working
As you mentioned, and already know, things can get finicky when batteries start to get low’ish. I would start with replacing batteries all around (which it seems you did)
I think you are on the right path with removing the tail lights to eliminate any thoughts about interference between things. I’d look back at your maintenance log and anything electrical on it try and isolate it to verify if it helps or effects the issues.
Especially with intermittent issues I like to verify things multiple times. I see many people take something off, unplug, remove or change a setting and then go “that was it” or “that didn’t work” 5 minutes later. I like to run it for a bit (maybe a few days) to see what the results or outcomes are, and then even put it back to the way you have it not to verify that the issues come back again. Isolating and changing one thing or variable at a time helps a ton as well. Small steps
Next I would look into the doors and door handles. Most cars with the keyfob (even if they DONT have prox entry) still have antennas in the doors or door handles as the sheet metal and tub has a bit of a faraday cage effect on signals and electronics, so the manufacturers will have a small antenna receiver in the doors or door handles to help pickup where the key fobs are.
When the Jeeps working correctly and picking the fob up like it should, try to turn the ignition on and off multiple times and see if it will keep doing so while moving the keyfob out of the passenger compartment and outside of the tub/cab. Notice how it’ll stop picking it up usually when you cross the door threshold or out through the door window opening? I’m guessing it’s triangulating the position of the keyfob using the multiple antennas and this able to tell if it is a person in the seat trying to start it with the keyfob or not
I do remember people having trouble with the electrical harness with the JL’s and JT’s for the doors, something about when you pull them apart and plug them back together you’ll have some of the pins bend or push out the back side of the connector. (A Google search should lead you to posts from a forum). This of course can lead to all kinds of issues. From my experience with the antennas in the doors and handles, plus being a Jeep (Jet) that I know the doors have been on and off multiple times, I’m guessing, I can see this possibility being part of the problem as well. Or at least something to check out.
Lastly, on other push button start vehicles, I know in an emergency when your keyfob batteries die you are still able to start the vehicle by holding the keyfob upto the push button or right on the ignition column and it should recognize it. From what I gather it was something about the board in the fob, when powered by a battery ofc, would amplify the chips signal for the vehicle to pickup. But when dead, in an emergency, you could still hold it up and use that to start the vehicle. I’ve seen this in multiple owners manuals across diff brands
Hope some of the helps Eddie!