What in the Hell?
 

I've replaced the postive battery cable. (The negative is in perfect condition.) I've replaced the battery. I've replaced the alternator. The car still says "Check Charging System," still llights the battey idiot light, and still dies after driving a block.

What am I missing?

Only thing I can think of next is a ground fault. There should be a few grounds on the LS4.

alternator to starter, starter to battery, near passenger headlight, and should be a ground to the chassis.

Sounds dumb but is there power at the alternator ? The main power wire on the alternator has a fuseable link at the starter on gen 6 cars would imagine gen 7 is close to same . And orange wire on plug should also have power with key on beyond that it goes to PCM Once again this is info for a gen 6 but I don't think that they changed the alternator much .Good luck

Have you checked the voltage with the car running? Wouldn't be the 1st time ive seen a bad alternator be installed? Reason I ask , I had a sort of similar issue a while back with my 06 SS. My car sits for weeks sometimes months at a time not being driven. The last time I let it sit for about 4 months never touching it, and when I did go to back it out of the garage the thing was dead as a door nail. Tried charging battery to no avail, I guess it discharged to the point of no return. Went and got a new battery, put it in and the car fired right up. However the dreaded " check charging system" was lit up and wouldn't go away. I was showing 13.8v with the engine running and like 13.0v not running at the battery. My car wouldn't shut off after a block though, I could drive it anywhere. With the voltage showing 13.8 I knew the alternator was charging like it should. Turns out there was a bcm update that had to be preformed to address the check charging system message. Not sure if the car sitting so long and the battery discharging wiped something out or returned the bcm back to some sort of default setting or wtf. But had the bcm reflashed and never had the issue again.
I know this isn't your symptoms exactly but just sharing something I learned along the way.
With your car starting and then dying after a short while, id check the voltage and make sure the alternator is putting out correctly. It wont run but a short time on the battery itself as you know. Ive also seen the battery cables look fine visually but have corrosion down inside the sheathing you cant see an inch or 2 below where the terminals bolt to the battery. Chased my ass on that one time with a s10 blazer. changed battery,starter and other things, only to find the cable had rotted inside the sheathing, it looked fine to the eye but wasn't.

The guy who towed me last night also mentioned this. How is it possble for any of these to go bad? The alternator is attached to the bracket, the engine is attached to the subframe, etc.etc.

The ground near the headlight is definitely good, too.

Cars rust, the ends on the ground rust. Rust doesn't ground well. It's usually easy enough to unscrew the ground, clean it up, and tighten again.

So, I had a mechanic look at it today. He cleared the computer and got it to start and run. We started it again when he looked at some other stuff. The third time, it wouldn't turn over. He called a friend at the local Chevy dealer and they're convinced it's the ECM. It's not acting like an ECM. It's acting like a battery that's not getting charged, or not taking a charge. Except the battery and alternator are new.

Any ideas?

How many volts does the car have when it is running ? A alternator that is working properly will put out between 12.5 -14.5 volts if it doesn't have that when it is running it is Not charging. Have you checked the power leads at the alternator ? 3 Wires Big wire power at all times, orange wire power with key on, third wire goes to the ECM

Um...there are only two wires, the orange one and a black plug-in connector. I would have no idea how to measure the voltage on either of these without damaging them.