For argument's sake, lets postulate that the seafood was 75T instead of 53T yielding an overload of 22T.
The required speeds would be:
root (370 / 348) ~= 3% over those for MTOW. From Breel's numbers, I obtain:
Flap 20 -- Vr 166 V2 176
Flap 10 -- Vr 172 V2 184
Remember also that MTOWs for a given set of conditions assume an engine loss at V1 whereas here we had all engines "operating normally" instead of the nominal 75% power after V1.
Taking the reports attributed to TSB's Fowler of 130 kt achieved (don't know at what point) instead of the desired 160 kt (in conflict with Breel's and my numbers) and postulating a linear effect on the speed achieved (until hearing from the better qualified), the full thrust overload factor would have to be something like 25% which would produce an 87T overload and a TOW of 435T -- with that kind of overload (which I do not believe), I would think that one or more tires would fail during taxi or takeoff.
Reports seem to confirm that close to the full runway was used; so we seem to be left with: the parking brake left on, or a reduced thrust setting (reports of a suddenly increased thrust when well down the runway support this), or a combination of these factors, which then brings crew fatigue into the question.