The Flight engineer commented that although the brakes were hot and I would need some 25 to 30 minutes air cooling, i had enough energy to backtrack and commence an immediate take off.
Based upon your report I sympathise with you. Your decision was entirely within the realms of sound airmanship and the flight engineer was wrong in his suggested course of action.
To deliberately return to the beginning of the runway for an immediate take off when the brakes must have been extremely hot after a V1 minus 10 abort, would have been folly. The back-track taxi distance alone plus the distance of the following take off run would have further raised the temperature of the brakes and tyres to unknown temperatures. Then to deliberately get airborne simply to cool the brakes and tyres in order to maintain a schedule, smacks of poor judgement by the engineer. There is more to good airmanship and sound command judgement than reading theoretical numbers from a book. It might be legal by the book but a decision to conduct an immediate take off, given the stated circumstances, would certainly not have been safe.
With notes against my record, endorsed because the flight engineer had advised me that it was safe
As you say, there were other factors that came into play but for the airline management to pin you because a FE said it was "safe" to take off again, smacks of personal animosity towards you. A good lawyer would soon destroy that airline management technical decision to castigate you for a command decision based upon your considerable experience as a captain. Flight engineers are quite entitled to give an opinion on what is safe and not safe but the law says the captain has sole responsibility for the safe conduct of the flight.