A good project management program does help immensely some custom injection molding companies follow this some don't. I would like to see the part is talking about sometimes the simpliest solution is the factor. I have been in plastics 38 years and you still learn on a daily basis.
and learn to take nothing for granted or assume everything is fine.
From the description I would guess the source is in the barrel or the plastic injection mold. I have seen issues where a dirty barrel can cause such issues. Material building on the inside walls off the barrel and burning and breaking off during mold cycling can cause this.
If the plastic injection mold has a hot-runner system and perhaps the T/C has been smashed and causing a reading from a different point than where it is supposed to, which will cause a zone to heat too high - but potentially read correctly. But you have ruled out the plastic injection mold (what about hot-runner controller or a gate valve system?).
It really sounds like a barrel issue, if it is the shortest path of material flow than it is probably near the nozzle. Have you tried changing the nozzle out completely and/or checking the accuracy of the nozzle T/C. Any bad heaters that may be causing other heaters to heat a section more than it would ordinarily (anywhere on the barrel but most likely on the nozzle or end of barrel closest to the plastic injection mold)?