I'm just posting an update in case it helps anyone else. After trying a number of different things that others have already tried, hoping I might experience different results, I finally installed the same 14 volt lithium battery Bradford_G and some others indicated they installed and resolved their system error issues. I installed a switch (tapped into the 12v, actually 15.5v, cigarette lighter/power point in the center console for power), ran the switch to an LC2i Pro, used signal sense on the LC2i Pro to turn on and send power to a Wavtech IRAD with 5 second delay, and the Wavtech IRAD turns on my JL Audio XD600/1v2 amp, which powers two JL Audio 10w3v3-4s in a ported box.
I didn't want to run power to the frunk due to not seeing any clear way to breach the firewall and due to the jump post on my 2023 Model 3 being totally different than the jump posts of prior Y and 3 models (posted two pictures of the jump post earlier in this thread). Bradford_G had suggested that maybe running power from the frunk to the amp worked for him due to the relatively long run of wire from frunk to the rear of the vehicle and enough voltage drop to prevent the car from sensing the influx of MOSFETs occurring when the amp turns on. I tested this theory in a very basic non-scientific way by connecting a 4 gauge power wire to the penthouse and simply bundling up 14 feet of the wire in the rear of the vehicle before connecting it to the 14 volt lithium battery I installed in the trunk. From the 14v battery to the amp I ran about 6 feet of 4 gauge wire. I have no idea if any of it makes sense, if I should have run equal lengths from the penthouse to the battery and from the battery to the amp, or if the long run from the penthouse to the battery was even necessary due to not trying a shorter run of wire first. All I know is that as it currently is, it all works. After countless hours trying other things that didn't work and constantly getting error codes on the car and killing the factory battery a number of times, I have decided to leave things as they are for the time being. I am sure I will eventually tinker with it some more and see if I can narrow down what the winning combination is. I'm just happy it all works and I didn't have to run a 4 gauge wire to the frunk.
The only issue I experienced after running the setup was after Level 1 charging at home for a few days and having no issues pulling a paltry 12 amps from my house, I drove the car to work and plugged into a Level 1 20 amp circuit. I then experienced an error code that said, "Power grid or vehicle issue limiting AC charging." While part of me wanted to believe it had nothing to do with the stereo system, I determined that it had to be the issue. After trying some basic troubleshooting to rule out the stereo system (e.g., disconnecting and reconnecting the charging adapter and charger, trying different charging outlets in the parking garage, and doing a basic system reboot), none of which worked, I simply disconnected the battery from under the hood and under the rear seat, flipped the circuit breakers I had running from the penthouse to the 14v battery and from the 14v battery to the amp, waited five minutes, reconnected the power under the rear seat and hood, and reset both circuit breakers. It worked and I haven't had any issues in over a week now.
I think the issue may have been due to the installation of the 14v battery. I charged the battery and observed the resting voltage to be 15.7 volts, which is .2-.3 volts above what the car's peak operating voltage typically is (15.4-15.5) when I installed the battery for the first time and connected it to the car. I think maybe the car sensed the higher voltage and thought there might be an issue. After installing the battery and running it for a while, the battery voltage normalized and was reduced to the same voltage as the OEM battery in the frunk (15.4-15.5). I would imagine that the car is incapable of sensing two separate 15.5 v sources at this point and simply sees a single source. This is pure speculation on my part, but it makes sense in my aging mind.
If I was going to do it all over again, I would first try running 14 feet or so of 4 gauge wire from the penthouse to the Wavetech IRAD with a 3-5 second delay, and the IRAD send the remote signal to turn on the amp. Maybe the relatively small amount of voltage drop and the delay (allowing the car to fully wake up and power all of its other systems) would work without installing the expensive 14v lithium battery. If that didn't work, I would go the 14v battery route but instead of running the long run I did from the penthouse to the 14v battery, I would only run as much wire as needed to connect the two points and would remove the IRAD from the equation but add in the LC2i Pro. If that didn't work, I would add the IRAD back in.
I hope this information helps others. I haven't ever posted anything in forums over the decades I have been using them to research issues, but after all of the time I spent on this project, and my wife constantly complaining about the time it consumed and being upset about the one time we took the car to get frozen yogurt and it throwing codes that we both thought meant we would have to Uber home, I thought it was worth sharing all of my bonehead mistakes with others to save them the time I wasted.