Poking around in Google Books gives references to la Baye d'Hudson from as early as 1730, substantially predating the Seven Years' War:
The reference is from Le grand dictionnaire geographique et critique by Bruzen de la Martinière. Note that that this passage also notes the French name as "Baye du Nord" (not "Mer du Nord"):
Les François la nomment la Baye du Nord, a cauſe qu'elle eſt au Septentrion de la nouvelle France, n'étant qu'à cent lieues de Quebec, & à autant du grand Lac des Hurons.
The French call it Baye du Nord, because it is to the north of New France, being only a hundred leagues from Quebec and as far from Lake Huron.
Whether or not there are earlier references to la Baie d'Hudson in the literature, I cannot say. But it does appear that in 1730, a Spanish geographer writing in French (de la Martinière was writing for Philip V of Spain) felt a need to use both Baye de Hudson and Baye du Nord in his work.
By the way: in that same work, la Mer du Nord appears to refer to the entire Western Atlantic north of the Equator!
MER DU NORD. On appelle ainſi la partie qui lave les côtes orientales de l'Amérique, depuis la ligne équinoxale au midi, jusqu'à la mer Glaciale au ſeptentrion. Elle a été ainſi appellée par contraſte, à cauſe que la met qui baigne le Pérou & la Nouvelle Espagne avoit été appellée la mer du Sud.
MER DU NORD. This is the name given to the part which washes the eastern shores of America, from the equator in the south to the frozen sea in the north. It was given this name as a contrast, because the sea that bathes Peru and New Spain had been named the Mer du Sud [Southern Sea].
Note, though, that this latter usage may not have been common in New France at the time (as noted above, this was a work written for the King of Spain.)