If you do that, the problem will shift to bouncing people around between two incompatible social and economic systems. "You are now subjects of Canada, until at some indeterminate point in the future, half of you will become Americans. Again."
The land these people live on is not of vital importance to either country. As such, we should be doing right by the people who live on it.
This may make Washington and Ottawa feel good (Because it lets both sides save face), but is going to be pretty shitty for the people living in said territory, if it ever goes back to being split.
If it will never be re-split, then it's a de-facto annexation. In which case, I'd prefer that the residents vote on it.
The land these people live on is not of vital importance to either country. As such, we should be doing right by the people who live on it.