Anglo-Americans had been trying to move west basically from as soon as they set foot in North America; there's a reason George Washington was in the area to touch off the 7YW, and it wasn't because he was sight-seeing. Likewise, the reason that essentially every treaty signed between the colonists (and later the US) and the natives fell apart was because the colonists kept violating the treaty and encroaching on territory that they had supposedly given up. You can draw whatever lines on a map you want back in London/Paris, it doesn't affect the people on the ground (since in practice neither the British nor the French had control of the region, nor particularly accurate maps thereof). People will settle, there will be conflicts with the natives, and a few photogenic massacres to inflame the local colonial government (or just particularly ambitious colonists) to put together a militia force and launch a "punitive" expedition. Fight a few battles, sign a treaty with the natives (who also have no reason to pay any attention to the lines drawn on a map somewhere in Europe), establish a fortified outpost that becomes a significant trade post, and then rinse and repeat, expanding as before. It's how things went before the 7YW, it's how things went after the 7YW, it's not like a military victory (which is almost certainly going to be less decisive than people seem to be envisaging, because the French don't have the numbers or capacity for a massive, overwhelming victory over the English colonies in North America; their strategy tended to be "fight to a draw in the colonies and make up for it with victories in Europe/India to win concessions that way") is going to change things too much.
As for the treaty "settling" things; keep in mind that the UK and France had found themselves at war every couple decades throughout the 1688-1815 period, and each of those wars had seen significant conflicts in North America. Even if the treaty says "absolutely no Brits west of the Appalachians" and the French and their allies somehow enforce it, that's going to be up for grabs the next time the British and French come to blows, and French forts are going to be an obvious target for British offensives when that happens.