A compromise settlement might be:
George III takes two new titles, “Emperor of the British Empire” and “King of British America.”
All British possessions in the Western Hemisphere, from Newfoundland to Barbados, are formed into a new Kingdom of British America, constitutionally separate from the UK, with its own American Parliament meeting in (probably) New York, and with a Viceroy there representing the King-Emperor – possibly one of his brothers or sons. The individual colonies’ established governments would continue; they would simply become provinces of British America.
A new Imperial Parliament and Imperial Government are established. The Imperial Parliament is supreme and sovereign over all kingdoms’ parliaments including the Westminster Parliament; it has representative members from all kingdoms and crown possessions of the Empire, including the UK, Ireland (still a constitutionally separate kingdom before 1800), British America, and the Isle of Man and the Channel Islands. The Imperial Government is responsible for all matters of commerce between the kingdoms, all foreign relations, all military matters (the Royal Navy and the British Army become the Imperial Navy and the Imperial Army – and are specifically directed to start recruiting men and commissioning officers from America), and administration of all colonies or possessions without kingdom status. Everything else is left to the kingdoms – but not on an “enumerated powers” basis as in the U.S. Constitution; simply as the way things are done, for practical purposes. Constitutionally speaking, there is to be no question that the Imperial Parliament has plenary sovereignty – including the abolition of slavery throughout the Empire, once that comes up.
That would have left the colonies with enough self-government to satisfy them, while still binding them firmly to the Empire – which, as a political unit, is limited enough in role that the Americans eventually effectively “running” the Empire would not be such a hard thing for the British – in this timeline, probably called the Old British – to swallow.
Having that autonomy, I expect America would experience – that is, the Americans would insist upon – the same manifest-destiny push to the west as in our timeline. Of course, it might mitigate that slightly – only slightly – if part of the settlement were that the Indian nations in British America get their own representatives in the American Parliament.
As for slavery, there would be abolitionists in the Imperial Parliament – from the UK and from the northern provinces of British America – and there would also be resistant members from the West Indies and the southern mainland provinces of BA. Once an abolitionist majority is reached – I do not expect the slaveholding provinces would even talk of secession, as there would have been no American Revolution to establish the legitimacy of the idea, no U.S. Constitution on which some theory of the right of secession could be argued, so secession would be purely and simply treason for the sake of slaveholding. Hard to defend.
In this timeline, Australia and New Zealand eventually would be added as kingdoms. Canada would always be part of British America.
The really interesting question would be how this constitutional arrangement would effect the Empire’s subsequent colonization of parts of the world such as Africa and India – for which autonomy or kingdom status would be off the table from the start, because, you know, there are already a lot of people there and they’re not white or Christian. Maybe 20th-Century nationalism there would take the form of demands, not for independence, but for kingdom status and representation.
Of course, it’s also debatable whether a British Empire-including-America would colonize Africa or India; the leaders might decide they have their hands entirely full with developing America.