They would have to take "Lower Louisiana", those lands part of the Louisiana Purchase , from Spain (originally French but offered to Spain after 7 Years War but still heavily influenced by French and would be given back to France in Napoleonic times) first. There is no other way for them to REACH inland America. The Hudson Bay Territory is too remote to maintain trade links and would presumably fall to America anyway in your scenario. With France and/or Spain in command of the Mississippi River and America in command of the St. Lawrence, the loyalist would have no outlet from "Upper Louisiana", those lands comprising the present American Midwest (eastern bank of the Mississippi, conquered in the 7 Years War from France).
I don't see a scenario where Great Britain loses all of her North American colonies from Quebec to Miami but somehow takes Louisiana.
The only possibility is that this is done by horse trading after the war. Say Britain conquers a Guadaloupe or Martinque or San Dominique (Haiti) and France offers something to Spain for Louisiana and then agrees to cede Louisiana to Great Britain to get it/them back.
France and Spain may like this idea as it creates a buffer between a rising American Republic with the Spanish Empire.
This may be your best option for something like this but I believe it may be unlikely.