So, I took it apart, and sure enough the main gear and the shaft showed the problem:
Probably the first "POP" I heard was one of the 10 gear teeth on the main gear start to buckle. At this point, the machine was still working, but I noticed that the gasket on the shaft had let some meat through. I shrugged at the time, but I think this is because the shaft was a bit looser now when it encountered this gear.
Then, when I started stuffing again, but with cold meat with less water than usual (1.6oz/lb) and at faster speed, this broke/bent weak tooth further. This is when I heard the "POP" the second time. At that point, I stopped working and the shaft moved up. However, the machine is programmed to move the shaft up at FULL SPEED. So when that happened, it began tearing every 10th tooth off the shaft because the gear tooth was bent and it was now moving at full speed. This is when metal fragments started dropping.
The guy at LEM very politely said that this is often due to stuffing without as much water, but also didn't try to fight on the warranty replacement or anything. He sent me a schematic and told me I could take it apart without voiding warranty or anything. I then texted him pictures of the broken parts and my purchase receipt.
Was it my fault for pushing it too hard, or was it a defect in the form of a weak gear tooth that buckled the first time under stress? I'm not sure, since it was a new unit, less tested, and there was only one tooth that failed (i.e., the other teeth could take the pressure). On the other hand, I was pushing it pretty hard...
In any case, it seemed to not matter to the LEM representative. It looks like he'll send me replacements, and I'll be good to go! Really good customer service. I'm so relieved!