The royal power in England was stronger than in France before the Magna Carta. England was the country who firstly developed institutions and tools of the royal state power. French capetiens did copy later many institutions and organizations created by Henry II Plantagenet.
You confuse two things there : royal power, and acceptance of this royal power.
During Philipp II reign, the royal power was far more accepted than it was during John rule. As Louis VII would be becoming king by the grace of barons and Magna Carta that could theorically limit his powers, the Capetians would have an hard time to make their power as accepted in England than it was in France.
So if the king of England is a frenchman, there will of course be resistance and unrest. But most barons in England were normans, that is normans from Normandy.
They were as much Normans of Normandy than Irish American are from Ireland.
It is quite easy to have barons remain loyal : you either obey or you will lose all your possessions. The capetians themselves had a long-established experience of defeating rebel barons on their royal domain.
The Capetians would have been kings of England because of a civil war made by England's great houses. Do you think they can afford to say "hey, fuck you, it's not as you can make another civil war against me"?
Louis VII is simply going to have the same situation in England that his ancestors had in France in the XI century.
I don't see why taking control of England would be at the expense of southern expansion. For this to happen, you would need reigning in England to be more costly than not holding it. This is rather counter-intuitive given England's ressources. When William the conqueror became king of England, this did not weaken his position in Normandy and in France, nor his ability to exert power and influence in France.
As far as I know, William the Conqueror didn't tried to double its territory in France, being quite too busy with keeping his english throne.
For England's ressources : a major cause of the war of English nobles refused military and financial request of John. They didn't wanted to pay for a war in France and aren't going to accept Louis VIII doing the same much more.
A good part of these ressources are likely going to be used in England for paying loyalties, secure royal presence as Capetians did in France (construction of castle, by exemple).
Personal union are maybe fun and instant in Paradox games, but in a more realistic course of events, there will be a cost.
You talk about being counter-intuitive if the conquest costed more than immediate benefit : as we're talking about southern expansion in Languedoc, it was actually more costing than anything at first for Montfort, then for Louis VII/Louis IX with the majority of the land standing in their pre-crusade owners.
Conquest was less about "woo, shinies" and financial budget than land, enforcing authority, preventing rival to appears.
And the total absence of mediterranean policy of Capetian kings (it appeared only during Louis IX, precisely because he had an access to sea, not the other way), it is more likely that ATL Louis VIII just make the choice of focusin on England.
But, okay, let's admit that he goes south. It's not going long before such a streched empire, from York to Narbonne began to crumble : the great curse of western medieval empire is simple, when you have an huge realm (HRE, by exemple) you'll have ruler ending by running to a corner to another, unable to fight a revolt 400 km south before the north began to rebel itself, draining ressources and finally forcing to abandon much more than you would have be forced to in first place.
So, I agree and reformulate : if Louis VIII or Louis IX gain some sense, they wouldn't try to expand in South.
Historically, Aragon's attempts to take control of the county of Toulouse was decisively defeated in 1213 at the battle of Muret.
Not at all : while Aragon was defeated in 1213, it keep supported financially and military languedocian nobles (while not directly).
The claims Aragon made were definitly (at the exception of Montpellier) by the Treaty of Corbiel in 1258.
Given how the Baron's Crusade against Albigensis failed OTL, leading Louis VII to finally intervene, seeing how Louis VIII is going to be busy enough with England, Aragon is going to have free hands on OTL Languedoc.
And you're forgetting that holding the crown of England and France together implies total control on Flanders and all what was going to become the (future Valois and Habsbourg) Netherlands, thanks to the legal and military power of the french king on the continent and to the wool of the english king.
No. Total control over Flanders would have meant that Kings of France and England would have Count of Flanders themelves : don't push the Capetian wank too much already.
And while Flanders enjoyed good economical and diplomatical (mostly because of the threat that representated the power growth of french kings with England, they're not going to be particularly pleased with their role of exclave when the Capetians would have more interest to develop wool industry on lands they actually control as Normandy.
Valois, Hapsburg rise to power is likely to be butterflied with such POD.