From what the vets I had the privelege to work with told me, the only thing one could do was throw a cleaning rod down the barrel forcefully and hope it either loosened the problem or forced the impacted round to fire upon contacting the pin. I heard fingers were lost due to carelessness.
on the origiinal M-16, the cross-carrier pin was configured like a cotter pin. The side of the carrier that the head belonged in had a deeply chamfered opening to insert the pin into, burying the looped end of the pin below the carrier O.D. The length of the pin kept it from protuding beyond the carrier on the opposite side.
With the loop end of the pin on the wrong side of the carrier, you could easily get the assembled carrier into the receiver, but it only fired once. This prob was widely known by the time I got there Jul 67, but those there earlier learned the hard way sometimes......some names are on the Wall due to this :(
The new carrier and pin fixed this, as you simply cannot get the carrier into the receiver if the pin is placed into the carrier from the wrong side.