Whilst I think it likely that George III would be moved be the arguements of leading colonials for more representation, and that he might even plan to introduce reforms, ultimately any such efforts will be neutered by parliment. The most that could be possibly made to stick are probably the creation of a peerage system for North America, and/or royal patronage for existing colonial assemblies. Neither of which are realistically going to solve the underlying issues.
It would likely however leave the King and colonists somewhat better predisposed to each other, and might have some interesting repercusions down the line. The king would probably be more receptive to further entreaties and though parliment would no doubt resist further, King George would be seen as a colonial ally rather than as an opponent.
This might paradoxically lead to a increase in support for the rebels, who would gain those who would otherwise have been loyalists but now view the conflcit though the frame of a benevolent and sympathetic king being thwarted by corrupt advisors. The king's support if it continues during the war, would probably lead to it being shorter as the British establishment's tolerance for setback would be even less and wider society, taking cues from thier sovereign, would be even more symapthetic to the American cause than OTL.
Another resulting quirk, especially with a large number of monarchists among the 'patriots' might be that the Americans decide to form a consitutional monarchy, just as many of the nationalist movements of the 19th century would later do, either offering the crown of America to George to be held in Personal Union, or to one of his sons. Quite possibly Prince Edward, which amusingly might then lead to Queen Victoria of America, who would then inherit the throne of Britian from her uncle.
In one timeline you might see the United Kingdom of America, and the United Kingdom of Britian in personal union for a singificant part of the 19th century, both developing as co-equal leaders of the larger British Empire. As technology and political thought develops into the 20th century, and the stresses of maintaining the Empire begin to take thier toll, it is very likely that sometime in that century there is either a final split that severs the two, or one more act of union that unites Parliament and Congress as one. Though by that point Congress would no doubt be the more powerful of the two institutions.