Harolds death at Stamford Bridge would have led to defeat for the Saxon forces there, due to the personal nature of leadership in the period. However, if the housecarls act as they did at Senlac Hill, and as they were conditioned to do by their society, they would stood and fought to the last, giving the Norweigen forces a battering as they died. This would suggest that Hardrads best tactice would have been to rest in Jorvik, regain strength and challenge William later.
However, Harolds forced march south was not a poor choice - the battlefield on Senlac Hill was an excellent position, and the Normans were bottled up next to the sea. Bar some bad luck - Gurth and Leofwine falling as the Saxon flanks swept forward on retreating Norman / Breton forces, thus checking the Saxon charge - Harold could easily have been victorious. Fighting the Norman forces in a less secure place north of London could well have spelled defeat even for the seasoned campaigner that Hardrada was.
The remaining nobles of England would also have to choose their new king in the witan, and I think that William had the stronger hold - the Confessors court had been a source of 'Normanisation' for England, while Hardrada was following a nebulous claim. This would have put at least the household troops of those nobles on the side of William, even if the fyrd could not be summoned again. This would have resulted in a gentler rule by the conquerer - no harrying of the north etc, and so a more secure Kingdom for William to use as a base against the Kings of France, the Princes of Wales and maybe further afield.