I thought I recalled previously strongly disagreeing on this subject with Faeelin, which is odd because I agree with every word (s)he just said. Interestingly, British attitudes viewing the Americans as actually Britons who wouldn't admit it continued long, long after American independence IOTL. There were British generals in the War of 1812 who refused to fight the Americans on the grounds that they wouldn't fight their own countrymen.
To those suggesting consolidation of the colonies into several 'Dominions' or a single 'Principality' or whichever, what I have to ask is: why? What motivation do the British have to merge colonies together? Just to be convergent to OTL? The last time they did it, it wasn't exactly a shining example of popularity; the Dominion of New England was a blatantly artificial, autocratic, profoundly unpopular creation of James II. That's a profoundly poor precedent.
And to those suggesting that the British would eventually realise they couldn't hold America and would hterefore have to give up the status quo and give more freedom to the Americans… again, why? That's just projecting modern-day conceptions—that if a subjugated people wish to be independent from an oppressive colonial government, ultimately they'll probably succeed—onto people who didn't have them. (Whether those conceptions are correct or not is completely irrelevant here; what matters is whether the British at the time would believe those conceptions to be correct.) In this scenario, the British have just experienced a major rebellion in British America that lasts around a year before being crushed. That's not very unusually successful on the scale of rebellions (there have been plenty of Irish rebellions much more impressive than that) and it's not going to convince them that British America can't be held without reforms. Instead, it will vindicate the American policy of the British government at the time. As Faeelin notes (I was unaware of that evidence but I'll quite happily use it), the American colonial assemblies are not going to be permitted to continue; any later British move to give autonomy to the Americans won't be a matter of giving more power to existing institutions, it'll be a matter of having to create democratic institutions from the ground up, against the fierce opposition of British American loyalists who have been administering British America more-or-less autocratically for the last decade or more.
The only way that comes to mind to maintain a stable British America ('British North America' as a term comes from after the revolution, when Great Britain had lost almost all of British America, as it was known, except the northernmost bits—hence the name) with a PoD after the beginning of the American Revolution is for the original factor restraining revolution to come back into existence: i.e. a powerful common enemy with a major presence in both Europe and North America. Great Britain had no shortage of powerful enemies in Europe and a distinct shortage of powerful allies there, mostly due to its own ill-considered actions. A coalition war against Great Britain—the same thing that was triggered by the American Revolution IOTL, but presumably started by some other cause ITTL—might lead to the defeat of Great Britain by France in Europe (IOTL the British got exceptionally lucky in that respect) and thus, perhaps, to France regaining what it lost in the Seven Years' War, not through any ability to project power to North America but purely through ability to project power to Great Britain and thus force the British to come to terms. Such a situation could also give the Spanish Florida and greatly weaken, perhaps even destroy, the British presence in the Caribbean. In that case, if the Native Americans whom Great Britain was attempting to court largely side against the British (presumably out of fear of American expansion against them), we might see an end to the Proclamation Line and a more vigorous pro-colonist attitude in London as a matter of national pride—which could bring British and American interests back into alignment. By that point the cozy pre-revolutionary status quo of 'benign neglect' is long-dead, but there might even be American representatives in Parliament. After all, a major cause of the expansion of Great Britain's suffrage IOTL was a matter of conservatives believing that the ordinary, poorer people would support their ideas more than the richer, liberal, currently-enfranchised people could (one of the greatest expansions of suffrage in British history was passed by ultra-conservatives for precisely this reason, though in that case for the cause of anti-Catholicism rather than anti-French nationalism), so if a Conservative government is in power in Great Britain, in that event—especially if it was a Whig government that was forced to give Canada back to France—the Conservatives might see American voters (though only the American property-owning classes, of course—not that radical!) as likely to vote for the Conservative Party's policies: focusing on Great Britain's worldwide colonial empire instead of Europe and taking a hard line against the evils of France, Spain and Roman Catholicism.
The biggest difficulty with that scenario, of course, is that in any such coalition war it'll be difficult to prevent the Americans from rebelling again and establishing independence with the help of France et al. Admittedly it's not the most plausible of scenarios, not by a long shot, but it's the best I can do if we require a PoD in 1776. (A PoD before 1776 makes things much easier.)