Couple of questions.

When you pressed the barrel in the trunnion, what gap did you set between the bolt and the carrier.  Hence with the bolt locked home under full spring/running home pressure to make sure that the bolt is fully locked home, you should end up with a .020" gap between the back of the bolt, and the front of the carrier once the upper has been shot for a few rounds.
Note in spec is .015"~.025" on the 223.

[youtube]https://youtu.be/y7izvL3E40k[/youtube]

On the barrel, is this a new after market barrel, or one that came with the kit. Hence the fluting in the chamber does not look ideal, and in fact the fluting mark in-printing on the spent case is less than ideal as well.   If a aftermarket barrel, then you may need to go in with a pick to clean up the fluting channels, and make sure that they are all solid from the end of chamber cut and through the two steps of the chamber as well.

Just a FYI,  but looks like the fluting on the chamber at the shoulder to case walls edges looks a little shallow, and why the flute marks are not deep on the side case walls far enough back.

I ran into this in the past on some aftermarket 91 barrels, and the way I solve it was to go into the chamber flutes with a small round Jewelers file with sharp tip point to ride the flutes inwards into the chamber and to deepen the flutes at these corner areas with just pushing forward plunge cuts with the file tip down the flutes.  Hence the flutes where EDM cut, but a reamer was used by hand in the end to clean up the flute to chamber edges/bring the chamber to final dimension, and this can cause a problem with the flute depths at the main case to shoulder flute channel depths ending up too shallow in the end.
Note, the way HK produces barrels is hammer forging, with the flute channels part of the  chamber and bore rifling mandrel instead. On after market barrel that are just button produced, they will chamber the barrel about .001" 'ish short, EDM cut in the flutes into the chamber, then use the reamer for the last .001"ish to leave the chamber walls/end of flute channels smooth in the end.  Hence it when the chamber is left a touch too long for EDM cutting and then final clean up reaming, then it causes the flutes to be shallow in the corner edges.

If you look into the chamber with a flash light, it's easy to pick up where the flute channels go shallow at the step edges and the file trick works well to deepen the flutes back up at these step edges.

The depth of what the flutes should be all the way from the back of the chamber, all the way to the end neck of the chamber, as well the two angle steps as well.