Re Hi-bypass engines - If N1 is used to establish TO thrust, then with bleed off you are running lower EGT for a given N1. So with bleed off (or A/C packs off on 747 classic - see note) you can set higher N1 > more thrust > more TOGW. (It is assumed the bulk of the thrust comes from fan flow, so any change in core thrust is not significant overall)
But the APU TOGW increment I mentioned earlier applied in either "packs on or off" cases.
(Note - bleed air drives hydraulics on the 747 classic - can't speak for the 744!)