Have the US provide aid to the Metis in their conflicts against Canada either during the Red River Rebellion, the the North-West Rebellion, or some other conflict of your creation. Then include consequences of the aid provided.
The Dominion government will have to exert itself more vigorously in suppressing the rebellion but it will still be suppressed. There will be no need for British regulars. All this is going to do is sour relations between Washington and Ottawa/London. Direct intervention is out of the question as that will not end well. Britain is still in as position at this point to defeat the Americans, but it is going to be costly to both parties.
Britain and Ottawa are hardly going to see such intervention in their affairs as a positive development and will take punitive action when they finally win. They could very well respond in kind regarding the Sioux.
Even if the Sioux were eventually defeated, they will almost certainly be given safe haven in the house of the "Great White Grandmother".