Did you ever go to the Performance manual and verify or refute the engineer's statement? I can't imagine why you would not...
Whether the FE's Performance Manual calculations was correct or not doesn't really matter in this case. The captain weighed up the facts of the incident and exercised his judgement to be on the safe side. If his company management elected to hang him out to dry then their agenda needs to be examined. A half decent lawyer could have provided a legal argument that the captain's decision not to take the advice of the flight engineer was perfectly acceptable and on the face of things, the safest option.
It was considered by the prosecution team that it was unlikely a double jeopardy event would occur such as an engine failing on take off following the hot brakes departure, requiring the landing gear to be retracted to maintain a safe climb gradient. In other words they felt it could be ignored in terms of risk assessment.
The problem with that theory is if a take off is made with known hot tyres and brakes there is a better than even chance that one or more tyres may be damaged by the excessive heat. If we accept that possibility - and a wise pilot should - consideration of the danger posed by debris from a deflated hot tyre tyre that may impact flaps or be ingested, needs to be taken into account.