Money has been mentioned, it may not be as bad as it seems. Henry now has twice as much land to tax. If he can give some kind of, 'tax brakes' to the new French he could be quids in.
One thing that might a bit of a sticky wicket is the church. Who would be in charge? English church in England and French church in France? Or would the be a power fight? And could his title be an issue? King of England and France or King of France and England? Edward III had a problem with that.
Money was always a major issue for Henry. He refused to tax the French, and the English were unwilling to fund him even when he was successful. Heck, if I recall correctly, the time he returned to England in 1421 and got Catherine de Valois pregnant with Henry VI he originally returned solely because he heard that in his absence Parliament had grown bold and had cut off most of his subsidies and were refusing to send the troops that they'd promised to. This was less than a year after he'd gotten himself recognised as heir to the French throne. Just because he was doing well did not mean that the English were willing to keep funding him - in fact as he did better, they became more stringent and tried harder to force him to end the war by strangling his money supply...
Edit: forgot to address the Church bit. The Church in Paris was independent of England, and always would be. Rheims had an Archbishop, and so did Rouen. Both of these were under English control IIRC - Rheims because they'd captured the Archbishop and Rouen because the Archbishop had actually defected, hoping to play the system and win power in Normandy. There won't be a power struggle.
The only lands directly under the control of Charles and his party is Bourbon Armagnac, Orleans and the parts of Gascony and Guyenne that had traditionally been English but were under their control.
You're forgetting the Languedoc and the private estate of the Dauphins - Dauphine - which together probably made up half of what Charles VII had left under him - so that instantly makes him a bigger landowner than all the rest of his vassals put together, which kind of turns your point on its head. The Languedoc committed a fairly substantial part of his tax revenue too, so he's got a stable, if diminshed, source of income there too...
I fully agree that Henry V faced horrific fiscal problems. But so did the Dauphin, so did basically every medieval King, bankruptcy was the norm and as that map you so nicely provided shows Henry V has more people to tax. He can't squeeze Normandy has hard has the Dauphin can squeeze Armagnac but then Henry V doesn't have to worry about he core tax base being conquered. While Henry V is going to have to periodically go onto the defensive due to financial problems that when you launch a chauvache, a self funding way of trashing the Dauphin's tax base. Though it does rather clash with the whole making yourself loved thing so it's a tactic to use sparingly.
This is indeed the problem. Henry refused to tax any French lands - even the traditional English "de jure" lands of Normandy and Gascony - because he wanted to win the French over to his side and, well, provide them as little incentive as possible to rebel. That means that the English alone are funding the entire military campaign of Henry V. That isn't going to change. As I mentioned before, the Burgundians were doing most of the actual garrisoning of northern France but they were starting to hurt financially too and Duke Philip wanted Henry to take over much of the responsibility - something Henry was literally incapable of accepting by way of not having the men to spare on garrison duty.
Whether Henry could conquer the south is not just a question of whether Charles VII would flee or whether he could win some more battles - it's a question of where he's going to get the men to keep marching south from, and the money to pay them, too.