Edit (questions)
1. When you charge it for the first time and the bolt doesn't go fully forward, once you've mashed the forward assist a bit is the hammer still cocked and ready to fire? Does the same issue happen when firing or just on the first shot?
2. If you
charge it on semi then switch to full auto and fire do you have any issues?
3. If you charge it on semi then
switch to full auto and cycle the bolt again, do you have any issues?
4. Please raise the hammer (i.e. pull the trigger) and then point your camera down into the lower and look towards the part I've put an arrow indicating (keep reading and you'll see why that bit is important) as I need to see the bottom of it as it may be deformed:
Nothing appears out of the ordinary from your pics, but obviously the trigger sears and the bottom of the full auto sear are concealed.
It's very hard to diagnose such a problem without being able to see the sears (but that means a full disassembly), however I've seen this happen in an LM4 before (functionally identical design). The way the full auto sear works (bit on the left in that latest image you showed me) is it holds back the hammer slightly as the bolt reciprocates then allows it to drop again. If you're interested in the motion, then look up a real M4 as the mechanics are the same and indeed the full auto sear is identical.
Problem is, the hammer is meant to catch on the bottom of that there sear. You'll notice when you switch from semi to auto it rocks backwards in such a way that the bottom of that sear is now in the way of where the hammer would be during normal cycling. Like I was saying earlier, if you pull the trigger and reset the hammer manually you'll see it sits slightly differently in the lower than in semi.
All in all, that full auto sear takes a hit every time the bolt cycles as it holds the hammer back slightly so that it isn't released before the bolt carrier unit is back forward again. Basically, the stock ones get destroyed all the time. I'd advise removing the full auto sear to check the bottom (don't lose the C-clip that you have to remove from the pin to push it out). In the image below, it's circled in red (that's the bit that gets kicked a little every time you fire in full auto):
There's no other explanation for it really other than something seriously wrong with the formation of the hammer or bolt. I would advise getting an aftermarket one simply for longevity if you're using an HPA-tapped magazine, but I can't find one for the LM4 series - just for WA-based M4s. The WA ones should be fine, so I'll find the cheap steel one I bought if it turns out to be the problem.
Here's a video on how to get that sear back in: