Another alternative way for a Saxon victory would be for Harold to charge with the Huscarls immediatly after the repulse of the initial norman attack. A full-fledged counterattack, instead of the dissorderly mob of fyrd who left the shield wall, could probably have routed the Normans, especially if they can dispatch William.
Agreed that, by the time Harold fell, the English were a lost cause for the day. Harold could possibly retire with the core of his forces intact, and replace a fair portion of the lost fyrd with new levies (if nothing else, recalling those soldiers who had been with him at Stamford Bridge), and offer battle at some later point in the campaign. But the very best that he can hope to achieve at Hastings with this late a POD is a bloody stalemate, and even that probably forces him to withdraw. In all likelyhood, William is going to come away from the battle bloodied, but in control of the south.