You do not invade another nation's mainland if you do not want to poison relations. Say, somehow Britain gains a foothold somewhere along the eastern coast.
What to do afterwards ?
Expand and try to gain some objectives that move you towards victory? This option requires a long and hard slog that will not end in a single campaign season and would require a bucket-load of resources to finish. Why would the invading army believe that in a long war the defender would not seek to use its local supremacy of logistics and manpower to beat them ?
If you suppose the war is going to be so insignificant that a negotiated end is possible, then why launch an invasion in the first place ?
Once you invade, you do not get to say where the finish line is, the Americans could simply refuse because you have not caused them such an injury that they should seek a conclusion to the war.
What would be the peace treaty in such a war anyway ? No American government would ever squander away its home advantage to accept such a loss, and the hypothetical government that did so would be voted out and a belligerent one brought in.
If a war was over small issues then it is bound to be colonial, and a colonial war does not require a response that could bankrupt the invading nation.
What about the long term prospects of relations ? US is not going anywhere and neither is Canada (here it is assumed that GB invades).