To answer at more length, the Dragoons did alow Hanse to focus the war on that front into a smaller area, and alow him to send more units and more resorces to the Cappie front. That this would go down was I think clear to all sides. Takashi had sworn a blood feud, and Jamie Wolf as a Clanner and Takashi as a samurai take thouse sorts of things very seriously. Hanse knew that (well, half of it) and was able to plan accordingly.
Had it been otherwise, had the Dragoons lost on Misery or had Theodore taken over right then or had the Dragoons just peacefuly parted ways with the Dragon, the resualt would probably have been a less agressive push into the CC, and a greater need to hold troops on the DC front. This probably means fewer worlds taken in each wave against the CC, meaning slower progress and a larger CC when the war finaly ended. The over all stratigy would have of course still been effective, just on a smaller scale. There might have also been some greater losses on the DC front, but I think Hanse was prepaired for that and willing to accept it; it would only have made him more gung ho for the war of 39, which may have come even sooner, if over the objections of Kat Steiner.
I think the greatest chance to really change the war is if the Dragoons don't mauld the DCMS on their way out. In the years leading up to the 4th Succession War, the Dragon managned to turn out regements at a frantic pace, with both the Ryuken and Genyosha comming on line. With out Misery, the Ryuken regements would have been alive and free to duke it out with the proto FedCom. The presence of an extra several regements of skilled troops would have made a larger difrence, I think, on the whole. Its possible that Hanse would not even have contemplated his bold attack if the Dragon had gained so much strength, and not lost it.