I would think the best way would be for a longer, more drawn out War of 1812.
American ships were raiding England, and if they burned down a few monasteries and towns that would do the trick.

Also have a better Quebec campaign that resutls is America taking the territory.
As for America, have the British burn Washington more than once, and also more coastal cities such as Boston and New York. If they capture and execute the President as a "traitor" that might heat the fire a LOT. Even better, have them force America into a position similar to that of OTL Canada. Of course, as soon as America rebuilds it promptly declares complete independence and reinstates the old contitution.
Some interesting side effects of this would be the fact that slavery could have begun a decline while the British held America hostage making the ACW less severe or butterflying it away completely.
If you want to keep the rivalry fresh, have the British declare this a "revolution," and voila, the Second American Revolution is fought. America wins of course, this time it is better prepared.
Over the course of history have major or minor wars between the USA and the Empire, possibly even have the Empire openly support the CSA is the ACW happens.
You woudn't even need that, though. The SAR will be enough.