Not impossible, just unlikely.
France never showed any interest in or inclination to ship tens of thousands of colonists to North America. There's no reason to think this would change because of a different 7YW.
Conversely, it is entirely possible, because it happened post war. The French tried to send around 13-14,000 settlers to French Guiana after the war, immediately afterwards in fact in 1763-1765, largely in response to interest in English style settler colonialism. This was a failure due to diseases on the ground (one author tries to claim that it was only due to bad planning and it would have worked out splendidly otherwise, but I have my... severe doubts, about that hypothesis), but the settler colonialism impulse is not only possible, but indeed historically happened.
Presuming the French win the Seven Year's War, the problems with France's North American strategy will still probably be exposed, as the severe population deficit will have led to severe difficulties resisting English offensives in this theatre. It is not only possibly, but indeed probably, that just as historically the French will come to the conclusion that they need to found settler colonies of their own to compete with the English. The main difference from OTL is that they actually have colonies where they can do that.
With what army?
France had a few scattered forts (which doubled as trading posts) in the region (and not "along the border," which was usually more or less a hypothetical construct anyway). The Appalachians (or wherever the border ends up drawn ITTL) are a huge region, and more or less impossible to police, especially since it's more or less occupied by various native groups (as opposed to the French) anyway. The British were unable to keep the Americans from crossing over after the war, and I'm not sure why the French (who are much further from their main bases) would do any better.
Just use the native Americans and some French troops to form a picket against the British colonists. The British hadn't been able to restrict their own people from moving across, but I doubt that the British tried as hard as the French would.... to the French, it isn't a diplomatic nuisance, it is an invasion, and the French response in movement into their territory could be a military one, unlike the British who gradually extended their lines of settlement West and never did much to actually stop the colonists.
As for sending over tens of thousands of colonists a year: sure, in theory it could be done, but in practice? That would be a hugely expensive undertaking (and the French finances, even in a victorious 7YW TL, were not exactly in great shape) and require major changes in colonial administration (not to mention almost certainly anger the natives, as those colonists get up to the same sort of shenanigans the English colonists did). Indeed, one of the reasons for the sad shape of French finances was the
utter debacle that had been the last major French attempt to massively invest in Louisiana (which had seen a few hundred immigrate, as opposed to the tens of thousands that you are suggesting).
While I've never seen any document estimating the cost of colonial expeditions as a whole, wikipedia lists the British First Fleet to Australia as costing £84,000 to outfit and dispatch, the equivalent of £9.6 million as of 2015. While there were doubtless other expenses, I can't imagine that they drove the costs up
that much, as compared to the cost of, well, sending a fleet of 11 ships and 1,000 to 1,500 people to Australia. If we assume the French send the equivalent number of settlers that the British sent to their North American colonies per year in the 1760-1765 period, which was 9,200, then that is around 6-9 more than was sent to Australia. Therefor, assuming that costs hold the same, that is only around £57.6 to 86.4 million in present sums for the French to send the same number in a state-backed project. It would presumably be significantly less actually, since North America is closer and they're shipping their settlers to established colonial territory. Let's say 40 million. I'm not sure where to convert British pounds of the era to French livres, but that isn't a huge expense.
The Mississippi Company itself was not the cause of the French financial situation, but rather a result. Nothing in wikipedia's article on it suggests that it was an especially expensive undertaking; conversely, it was its failure to fix the French financial situation which gained it notoriety.
Disease, problems with natives, and government on the ground is more problematic, but I would suggest that these issues are over-estimated. The French have a vast amount of territory in which they can potentially settle, and it wouldn't be that difficult to go to one region, accept that the natives there are lost to France, and continue friendly relations with other Indians - like the Indians on the East side of the Mississippi, who can make a useful buffer against the British, supported by French troops and weapons. Similarly, the French have a vast amount of terrain; not everybody needs to be settled into the Louisiana bayous or even modern Louisiana in general, they could go to Quebec or to the north of Louisiana.