PCM replaced and Fuel injector not working after reversing the battery connection?

No, I wasnt saying it was a throttle body issue, sorry I should have been more clear. I was just showing the difference between the California and Federal emissions standards for this vehicle. In the first case (Cali Emissions) that PCM wire (PCM pin 1AY) is power coming from the Ignition switch to the PCM.
In the 2nd case, (Federal Emissions) that same PCM wire (1AY) is Grounding the relay that powers the Drive by wire system.
So the difference is that with the key On, the California emissions vehicle will have battery power on 1AY. The other Fed emissions, with the key On, should be Grounding that circuit and therefore read 0volts.
Putting on a different throttle body will most likely require an idle relearn procedure to be done.
Here is the resistance test for the throttle body (1st diagram) if you want to verify that.
And the 2nd diagram is for checking for an Open or Short circuit from the throttle body to the PCM connector.
It might be that the throttle body is shorting out, or going open circuit when it gets hot. Circuit board issues will usually show themselves under different temperatures. Since you have already tried different PCMs with the same fault.

Does the vehicle have a sticker under the hood that reads either California or Federal Emissions on it? So we can figure out what you should be reading on that PCM wire.

Okay, some of these wiring diagrams are incorrect, you're getting power on the green/blue wire, correct? The OEM wiring diagrams show different wire locations, the 1st diagram below is labelled as "Except Cali Emissions" and has 1AY stating Less than 1volt with IG Switch On. And the 2nd diagram is the opposite. So, I think they have them incorrect or some other option might be available, but since you have tried different PCMs, and still have the fault, it's obviously not the PCM.
But the other throttle body does not respond if you put the vehicle in Drive? But no longer stalling out? Is it setting any different codes with the other throttle body?

Check the fuses that I have marked in the yellow squares for any corrosion or signs of the fuse having darker tabs on it, those fuses feed the PCM to and from the Main Relay. The P0606 code is mainly caused by a loss or lower voltage to the PCM. A fuse doesn't have to be blown to have voltage drop on it.

Okay, so the issue is now showing up when you turn the Radio on, that's very interesting because it looks like the Radio is on the Medium Speed Canbus network, and module faults from either shorting out the network, or sending out corrupt data packets can cause other modules to malfunction. Especially if you have a display that the radio is tied into, and possibly a navigation system, these will all talk to each other on the vehicles communications networks, High Speed, Medium Speed canbus, etc.

So, it looks like the Instrument Cluster is the Gateway type module where the 2 different networks will communicate with each other through, the PCM (engine computer, TCM trans computer) are on the High Speed network, and the Radio, Climate controls, information display are on the Medium Speed network, so the radio might be interfering with the entire data bus in the vehicle. I've marked the High Speed with Red arrows and the medium speed with blue arrows to show how they meet up. The Passenger Junction Box is also on the medium speed network, so it's possible the Radio is taking down the Junction Box which houses all the fuses and relays inside the vehicle.

See if keeping the Radio off for a while affects the stalling issue.

Are you getting any codes setting in any modules at all? Sometimes data from modules that stop communicating or are setting certain fault codes, other modules will default to known good data and stop using corrupt data from the bad module, this happens in some GM setups, but does depend on how everything was programmed.

If the AT warning lamp was on, it should have a trouble code stored in the Transmission control module. You will just need a scan tool that is capable of scanning other modules besides just the engine computer. So a full system scan with the Cluster plugged in, might give you other codes to give some direction as to what is faulting,
Since know that all these issues started when the battery was hooked up backwards, its pretty safe to assume that this is an electrical/module issue, a lot of damage can occur when a battery is accidently hooked up in reverse. Theres some info on "Stalling when stopped/ with AT warning lamp on" and service info actually gives the voltage values of what you should see on each wire of the Transmission Control module, Ill post this for you if you want to go through and check the voltage readings at the TCM.

These are some of the diagnostic steps for checking the voltage values of each transmission control module wires at its connector. The rest of the troubleshooting section of this is all oscilloscope waveforms, and gets pretty technical if you don't have any experience using a scope. 10-14 are some examples of scope waveforms that certain sensors inside the transmission should be inputting into the TCM.
But if you want to check the voltage values with a multimeter, it will be DC volts using the Signal labelled "GND" for your Ground connection.

The voltage testing at the TCM connector is not as difficult as it looks, you're really just probing the back of the connector with a multimeter and looking for the voltage level specified under the "value" column. But with this P0606 code popping up again, the vehicle might be losing its learned value for the throttle body (idle relearn) if theres a voltage loss occurring somewhere.

Because the U0100 code is a communications error with the TCM that looks to be coming from the PCM, so it's having an issue with talking to the transmission control module. And if the TCM is either losing power (which is usually the cause of a P0606 error) that means the throttle body relearn may also be lost and that is causing your stalling issue.

Since you have tried multiple PCMs with the same result, my thinking is now heading towards a damaged TCM due to the battery accident, in many cases the TCM will also use the throttle position data from the PCM to control the pressure in the transmission for different shift points.

Some times I like to power up a module on the bench and check for any communications voltage coming out, since it will be disconnected from the rest of the vehicle's network. Because it can be difficult to de-pin the Canbus wires from module connectors if you're not used to doing it, the TCM looks to have 3 power inputs (12v) and 3 Grounds. So let me know if I can do anything else to help here. Id really like to see you get this thing fixed, so you dont have to deal with these issues anymore.
What type of scan tool are you using currently?

The TCM is located right next to the PCM in the engine compartment, are you located in an area where getting parts is a challenge?

Are you still getting the same codes setting? With the key off, check the TCM connector, see if there's any corrosion or pins that look like they might have overheated. If you think about things improving when the vehicle warms up, circuit boards and connectors will expand when they heat up, so even a loose pin in a connector might make better contact when it gets up to temperature. The fact that they keep the TCM and other modules in the engine compartment is just not a good design, being exposed to the elements and various temperature changes. Thats one of the reasons I like to inspect connectors in locations where they can get corroded, corrosion shorts out pins together and it doesn't take much to do it. Just things to keep in mind.

This post is a few months old, but replacing the harness connectors for those sensors and the throttle body took care of a communications code, it looks like you were having? That sounds like a 5volt reference circuit was being shorted out and taking down a module on the network. Glad its fixed though, always good to hear.