He would no doubt have gone on the face his Norman cousins under William.
This is actually a bit debatale : while the ties between the courts of Norway and Normandy were effectively meager at this point,
there were still an acknowledgement from Normandy's role for Norses (less due to a really thin Anglo-Scandinavian heritage in Normandy, that Normans played a key role into Christianisation of Norway).
It's is quite possible that part of Norway's bishops were consecrated in Normandy in the decades before the Norman conquest of England, and it have been hypothesized that the events of 1066 were based on a roughly joint attack on Harold, with good reasons IMO
On this perspective, we could see Danes and Normans splitting up England in a North-South division reminiscent of other shared kingship in the Xth and XIth centuries. How it would unfold from there is anyne's guess, altough the unstability of Scandinavian transoceanic kingship would make me think that Anglo-Normans would have better chances winning it all eventually.
What do you guys think would have happened in a battle between these two Viking descendants? How would their fighting styles have compared?
At this point Normans were essentially French : not only due to a Scandinavian settlement (
or rather, a mix of Anglo-Danish and Hiberno-Norse settlement, more than Vikings in the sense of settled raiding bands) being fairly limited to the coastline; but also culturally as they were in relations and eventually mixed with Neustrian nobility even before the treaty of 911.
In 1066, Normans considered themselves as Franci, French. Of course, a special regional kind of Franci and not as a proto-national ground but that's true of virtually any region of France at this point too.
There were simply no real interest from Norman court to Scandinavia after (and probably during) Rollo's reign and the last skald visiting Normandy in the 1020's was essentially about an aged duke's remembrances at best.
When it comes to warfare, Norman fighting was as "french" (or, really, continental) you could get in Champagne, Anjou or Flanders.
Now, let's imagine a later conflict between Norses and Normans.
I mentioned it above, but "North Sea empires" à la Canute's aren't really stable and actually to crumble easily under their own weight : not only because regional elites had their own political culture and interests on which kings had to stress their authority to mobilize enough forces and resources to defeat local/regional uprising or kingship claimants which generally tended to prove too much after a while.
It's of course complexified by a certain tendency of scandinavian kingship to aggregate titles not as much dynastically but as themselves : roughly, a Dane king without relation to his predecessor and even managing to stress its independence towards the "regular" successor of the hegemon would still consider the titles of the previous kings as amalgamated to his one.
Hence why both Norse and Dane king could claim the kingship of England regardless of their ties with the last scandinavian king of England.
Meanwhile, Normans benefited from ruling a fairly unified principality in the XIth century (especially compared to their neighbours), and a fairly populated one with that, with a lot of mercenaries/vassals/fighters to be found not only in Normandy but, as it happened in 1066, in Brittany and Flanders.
Giving the alleged ethnic hatred between Saxons and Normans should be nuanced (Normans didn't oust Saxon nobility because they were Saxons, they ousted them because they wanted their place and/or get rid of a traditionally rebellious nobility they already dealt with at home), it's not even a given that Saxons would join with Danes or Norses just because. Especially when a good part of the Saxon political identity in the Xth and XIth century was "not Scandinavian".
Honestly, strategically (and possibly tactically, even if it's really not a given), Normans have far better chances to take the whole of English kingship at this point, but as it would happen on very different grounds than IOTL, probably with important changes.