Chamber brush with CLP, used by hand will resolve that problems; and should be used every cleaning as well.

If it not fouling in the chamber, but a broken piece of a spent case, then single one piece cleaning rod down the muzzle, once it out the back side of the receiver, put a 30 cal brass brush on the end of the cleaning rod, pull the brush inside the split piece of brass up to around the neck section of the brass, then give the end of the cleaning rod a sharp blow to drive both the brass brush and spent split piece of brass out of the chamber.

Hence where you pull the brush inside the split case, the  brush bristles are going to angle back to get a good bite on the inside of the brass, and hold onto it as you drive the brush and the brass back out of the chamber (towards the back end of the receiver with just the upper off the lower and in hand alone).

The glitch as stated, no 300 black out spent case extractor,
As for the 7.62 broken case extractor, although the tip is right for the one Brownells sells, the back side is wrong to mate up with the bolt face.



Again, 30 cal brush is used for 223 and 300 black out cases, and 45 cal brush for 308 cases (ID of the main section of the case will determine the needed brush, since you want the bristles to be angled back just a few degrees).  If the brass brush did not work, then get a New 30cal SS brush instead.


And pull the brush back to bend the bristles towards the back of the receiver inside the  broken case without twisting the cleaning rod to the start of the neck of the case, and give the cleaning handle a hard single blow back down the muzzle so the bristles will bit into the inside of the case to drive the broken spent case out the back side of the receiver with the brush on the single piece cleaning rod tip.
Note, in a 223 case, you can just pull all the way back to the neck section, since the smaller neck of the 223 will not allow the brush to be pulled any farther forward towards the muzzle.

Not a fan of this with the barrel still on the upper (not enough room to work cleanly in the upper receiver),

Worst case and you can't get the spent brass out the conventional way with the brush back stoke, then time to pull the barrel off the upper receiver instead.  
Here, the pick method to get the brass out will work since you have plenty of room to work with; without fear of damaging the chamber side walls.  With the case/chamber being tapered, then it more of a exact method of tapping  the inside of the brass only, and not getting too deep with the tap treads to start cutting into the chamber walls instead.