I agree that piecemeal is probably the way to go.
The war of 1812 provides the opportunity to get New England via the Hartford convention, Tecumseh's confederation as an ally, and Louisiana as a shared British-Spanish condominium which would be de facto British.
Then I think you need to wait a while. Each of these areas needs to become more developed, and the British Empire needs to ban slavery. Britain then can wage a propaganda war against the remaining US, condemning the ongoing enslavement of people. You could imagine something similar to the fugitive slave act in the early 1850s pushing the remaining northern states to secede. These Free States of America, could then fall into the British orbit like New England before them. Maybe they could be bribed with Ontario, and maybe whatever is left of Tecumseh's confederation, to becoming a dominion as well. The rump USA might be willing to let them go to preserve slavery unhindered, they might even expand into Cuba or Haiti.
After that I think there's only one viable route forward, and it's pretty unthinkable for its time, but not strictly impossible. Britain and its dominions would need to train and educate a leadership class of African Americans in anticipation of an alt equivalent of the civil war. So after 1865 there would be radical reconstruction, Freedman bureaus on steroids, and possibly a fairly large Exodus of the most racist elements of formerly American society. the British trained African American leadership class would guide this reformed state, we can call it New Africa, into the British sphere.
Each of these dominions would have a certain level of investment in Britain. An imperial federation would certainly help too. By 1865 New England would have been a Dominion or something like it for about 50 years, fear of the US reconquering it keeping it with Britain for most of that time. Louisiana would have grown as a British colony from the very beginning, at least outside of New Orleans. The free states would view their triumph over slavery as a shared victory with the rest of Britain, but would be racist enough to not want to join with new Africa as a single dominion. For the same reason it might take a while to get new Africa into the imperial federation, but I think it's at least possible that the gratitude would last long enough for the racist attitudes to be overcome, especially as new Africa becomes an economically more enticing federation member.
This is my attempt anyway
