WI: William I becomes King of England with Anglo-Saxon support?

Here's a scenario:

Stamford Bridge results in a victory for the Norwegians, and Harold Godwinson falls in battle. Harald Hardrada emerges victorious from the battle, and subsequently occupies York before marching south.

Meanwhile, in London, the Witan has assembled to choose a new monarch following Harold's death. Initially, it is expected by most in England that they will chose Edgar Aetheling, yet instead the Witan offers the throne of England to William of Normandy (whose own invasion force has not yet set sail), on the condition that he crushes the Norwegian invasion in the North, and, upon securing control over his new Kingdoms, respects the rights and customs of the Anglo-Saxons.

William accepts, crosses over to England and advances North to meet Harald in battle. The two men, both claiming to be the true King of England, meet in battle, and the result is a victory for William and his forces. Harald is killed in the fighting, his armies enter a state of disarray, with many simply returning to Norway, and William is able to sweep up the rest of England.

Now William is King of England, as in OTL, yet with a crucial difference - he has the backing of the Anglo-Saxon populace, and cannot enforce Norman rule over England due to his prior agreement with the Witan. Therefore, the Anglo-Saxon lords (Edwin, Morcar, etc.) are still in place, the Witan will determine his successor to the throne, and it is likely in such a scenario that Anglo-Saxon culture (including Old English) will remain in place.

What would the impact of this be? How would William's reign unfold in such a scenario, and would the Witan chose one of William's sons to succeed him, or simply appoint a member of the House of Wessex to the throne (probably Edgar Aetheling)? How would knights and landless men in William's Norman Army react to these developments, having previously been promised land in England yet now receiving none?
 
OTL, William's army was largely raised by promises of loot and land in England. TTL, is he going to just lead the Anglosaxon fyrd north along with what is left of the housecarls and whatever Normans are still willing to come?

And how much does he actually respect the customs of England once he takes over? William was a masterful man in the old sense.
 
Is there anything necessarily preventing him from doing it anyway once Harald is defeated?
 
Edward the Confessor had a large number of Normans in his court. I suspect that William would just take the throne, bring across some of his most trusted friends (/cronies) and as titles became available, hand them out. Without the need to fight for his throne, he doesn't need the army. The Norman-isation of England will probably still happen, but much more slowly.
 
If Billy were to wed his son William to a high ranking daughter and get an heir from that then maybe the crown could be kept in the family. The question I want to ask is if a peaceful acceptance of the conqueror's rule would lead to a stronger position in England and on the continent. This could result in an earlier taking of Maine.
 
If Billy were to wed his son William to a high ranking daughter and get an heir from that then maybe the crown could be kept in the family.

Would it really be William Rufus who gets a high ranking daughter, rather than his elder brothers Robert Curthose (who admittedly was kinda left in Normandy IOTL, and might still be betrothed to someone in Maine) or Richard (who died young in a hunting accident IOTL)?

Are Gyrth and Leofwine dead alongside Harold? If they are alive then the Godwinson's are still a major force, and they might be miffed the Witan overlooked them when it came to choosing a new king- in such a scenario could one of Harold's daughters marry one of William's sons?

There's also the Atheling's sisters floating around as marriage prospects, and could be used to boost Norman legitimacy (especially if you can can contrive an accident for Edgar). But if they were born in the 1040s they might be a touch old for William's sons.
 
Would it really be William Rufus who gets a high ranking daughter, rather than his elder brothers Robert Curthose (who admittedly was kinda left in Normandy IOTL, and might still be betrothed to someone in Maine) or Richard (who died young in a hunting accident IOTL)?

I was just going with the one who made king, I don't recall the reason Curthouse was left as Duke, so I hedged my bets. Either son is a viable choice for the union depending on the bride, I was just going for the one who became King. Although, rather than the Duke of Normandy become the King of England, it could work the other way around, so that any King is the Duke. I'm not sure how one could achieve the result though.
 
IIRC William didn't seem to get on with his eldest son (Robert) but it would have set a bad precedent to disinherit him completely which is why he was left the Duchy. OTL The second son (Richard?) pre-deceased William, which is why William Rufus got England. Robert and William Rufus initially agreed to be each others heirs but then fell out, with a few minor wars between them (I can't remember exactly how Henry fitted in at this point - he just got money as an inheritance I think, initially, though of course he ended up as both King and Duke.
So, I can see William appointing Robert as his administrator in Normandy, with the intention of leaving it to him, leaving his betrothal as was (Maine?) and grooming another son - Richard or William - for the English throne. For safety's sake I expect he'd marry both of them off to English nobles' daughters. Henry might get a wife from either England or elsewhere - as the fourth son he'd be less important.
Separating England from Normandy would cause less friction with the French king too - he wasn't too happy with a vassal who was in a lot of ways more powerful than he was. Unless the two are re-united in the first generation (as OTL), I think they'd quickly go their separate ways dynastically.
 
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