Would Henry V be smart enough not to do this?
Henry didn't in the first place. He personally forbade the match to prevent the falling out of relations with Burgundy. Gloucester's response was to continue privately seeing Jacqueline, and then to marry her as soon as possible after Henry's death. In a world where Henry stayed alive, likely the marriage wouldn't happen at least for a good few years. However, I can't see England and Burgundy staying friends forever. It was no secret that the Dukes of Burgundy sought independence - at one point they almost persuaded a Holy Roman Emperor to crown them King of Burgundy, I think this was around 1430, and it was only avoided when the HRE got cold feet on the way there and turned around. Personally, in this scenario I see Charles the Bold continuing to support Henry in pacifying the north, then demanding independence in exchange. Henry would most likely reject this in the short-term, wanting to see Burgundy continue as a French vassal for a while while the north was properly brought under control. Likely this would then result in the Burgundians playing along until such a time as they could force the issue - maybe around 1430. I find it unlikely that they would defect to the French camp, as that would do their independence demands no good whatsoever. Rather, they'd likely just withdraw their forces from the garrisons and play hardball. However, Henry by this point would likely be in a position now where he could take over from the Burgundians. In this position, Burgundy would get its independence and Humphrey Duke of Gloucester would get his marriage to Jacqueline of Hainaut.
But then the issue becomes:
1) Why would Parliament not reject Henry V's demands? It wasn't just that he was dead, the country was broke.
2) Why was Northern France and Gascony an insufficient base for for the English, and how does Henry change that?
1 - England wasn't broke, it was just becoming broke. Henry had the personality to force Parliament to accept his terms, and he had the popularity among the citizens for them to want to keep fighting, just a little bit longer. If he decides to settle on consolidation, he can probably have his aims fulfilled in around 5 years, give or take a few, and this would just about be manageable. Remember that at the same time, the Langue d'Oc was also having massive fiscal problems, so it would be unlikely that Henry would need to constantly raise big armies to fight the Dauphin, he would merely need constant waves of reinforcements to take over the north.
2 - Because Normandy and Gascony were still somewhat opposed to him, or at least were wary of throwing their weight in either direction. As King of France he could likely bring them on board over time, there were a small lobby in France outside of Burgundy who supported his claim, but it would be a dangerous thing to send out the taxmen so soon. It would turn his new subjects against him and would risk partisan noblemen creating trouble by attacking those tax collectors, depriving him of money, and possibly create a new front Henry would need to fight on. Logic dictates that when you conquer a new area of land, especially a major one like northern France, you treat it like royalty for a little while, to win the loyalty there. Only then do you start using it to raise money.
Gascony is a bit more complicated, too. Gascony had a tradition of weak Ducal control and strong vassals. The Duke had little right to tax there at all, and the King of France had virtually no right to do anything there at all. Gascony was a traditionally troublesome region. The English were somewhat popular in the coastal regions because English trade had made the middle classes popular, and the English Dukes didn't break local laws and traditions like French Dukes did, which is largely what provoked the Gascons to send a plea to England for them to send an army in 1450 to retake control there when the French were finally mopping up, but it still really wasn't a great place for sponging money. There was only a small hinterland around three major cities - Bordeaux, Bayonne and Dax, I think - where the Duke was the local landlord and had right to gather taxes. For this reason, Gascony would only ever be able to raise enough money to fund its own garrisons, plus anything vassals might send out of pure generosity rather than loyalty.