The US does have more at stake than the British do in this war, this is true.
I suppose the real question remains, would the war go on for long or not? The British attacks in the Caribbean are likely to be early, although doing so will undoubtedly trigger a counter-invasion of Canada. Imagine Lee heading up the US Army's attack into Canada....
I believe the generalship available to American troops at the time was superior to that of the British at the time, given they were going through some major changes in how they developed officers after the debacles of the Crimean war just three years earlier. The Americans also have more manpower likely in theatre. And the Americans are likely to raise more of those troops over time.
While the British clearly have the largest navy of the time, the problem for them is it is the start of a major transition in naval technology. At first, the British will dominate on the seas, but I can see a determined US pushing the envelope on the emerging technologies and getting the industrial force of the nation to pump out ships within a year or so. Men like Dahlgren, Eads, and Ericsson can make things very uncomfortable for the RN if given time.
http://en.wikipedia.org/wiki/Ironclad
So the real question would be why wouldn't the Brits and US rather than dive into this madness too deeply, why wouldn't they settle on some equitable division of the territory, despite the protestations of the American Oregonians?
I suppose pig headedness leading to early bloodshed could be enough to enflame passions, and before too long both sides are too invested to easily come to the bargaining table....