Taking the comprehensive view:
1) In order to bolster the Southern states, you'd need British court cases to rule the other way on slavery, and therefore retard the development of British abolitionism.
2) I'm unsure you can butterfly a F&I war. While the specific war of OTL can be butterflied, the imperial conflict between British and French alliances, in North America, Europe and Asia, will still exist.
3) In this inevitable conflict, I am very doubtful as to France winning. Even if they win on the continent, the British have many more colonial soldiers and a much better navy to throw into the conflicts in Asia and the Americas, and I suspect that, at the least, the Ohio Valley would still be taken.
4) Unlike the Spanish viceroyalties and the Capitancy-Generals of Brasil, the British colonies, specifically New England, had had self-government for decades if not a century by 1776. While colonial governors had authority over them, you still had the significant development of a distinct American bourgeoisie with distinct and separate political institutions.
5) With British success abroad, there comes the need for taxes. Already the merchants were displeased with mercantilism, and the end of salutary neglect brought about a taxation regime that pissed off the colonists. Furthermore, mirroring the intra-US conflict between the gentry and settlers decades later, the British were much more respectful of natives and Frenchmen in the Ohio Valley than the American colonists were. People want to settle beyond the Appalachians, and to keep America British you'd need to allow settlement beyond the mountains.
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Ultimately, America was populous enough, independent-minded enough and taxed enough to lead to revolt. Any possibility to weaken Britain would have attracted French and Spanish attentions, even if France had Quebec. Furthermore, unlike sparsely-populated Canada and just-started-settlement-as-a-penal-colony Australia, America had the largest European colonial population anywhere at that time, further driving the need for economic independence and giving the Americans more manpower for a separatist struggle.
In my view, you'd probably need a British victory in the ARW followed by political concessions to keep America British, and even then I think it'd be a close-run thing. The Parliament and Britain needed America's economy bound to Britain, while America wanted more leeway in economic affairs.
I think that Britain could keep part of the Americas, as it did OTL. I think Britain could keep the Americas for a longer period of time. I am not sure, given that we saw Canadian rebellions in the 1830s and given America's long-standing trends of political independence, from VA House of Burgesses to Massachusetts, that Britain could have retained the 13 colonies, in full, forever. I think it could be done, but the various economic, military and geopolitical factors of the period give America a lot of reasons -- and opportunities -- to try for independence.