I think it would be interesting to think what would have happened if Britain kept the profitable carribean islands and gave New France back? Would this butterfly the ARW? With a french neighbour to the north? Also the income from the islands may mean taxes in the colonies might not need to be raised as much?
It is indeed interesting. As I've made clear above, my stance is that such a scenario is possible. The French are not in a position to dictate terms of the peace. The question of the OP is not whether such a scenario is likely. It is what happens if such a scenario unfolds. There are factions in Britain which, OTL, wanted a harsher peace. The British sugar barons successfully pressured gov't to spit out the French sugar islands. It is not outlandish to envision the harsh peace faction to win out. So, instead of blindly going down the AH path of derailing a thread by arguing whether the scenario can happen, let's discuss what unfolds if it does happen.
It's going to affect the sugar economy, but it needn't all be bad. the conventional wisdom is that the British sugar industry goes into a slump. But, world goods have a tendency to find their way into a void, and just as there'll be excess sugar in Britain, there'll be a void in French markets. Plus, just because Britain would look to punish/depress the French by keeping the islands, it doesn't mean that the new possessions are left on an even footing with existing British industry. The pain of the new situation may be shouldered primarily by the French (now British) sugar industry. My hunch is that the situation, while creating a disturbance, will not be dire or permanent. As stated, exploitation of these new possessions may alleviate the financial pressure on existing British colonies. My belief there is that those colonies, by that point, are already well on their way to looking for an excuse to go independent, and will likely revolt, regardless. Nothing is inevitable, though, so it is possible that the loyalists carry the day (although, as the saying goes, tomorrow is another day, and a future generation of rebels will likely look to gain the upper hand).
On the French side, a loss of revenue follows the loss of the islands. this impacts their ability to rebuild their military/navy as they look to prosecute a war of revenge. this may mean that they're a little more hesitant to jump in on the side of the NA patriots, and a failure to jump, IMO, means a failed American revolution.
You also have a lot of pissed off Frenchmen. their supply of sugar is interrupted. Eventually, the French/British sugar makes its way to them, but I'm guessing a layer or two of taxes has been added to it, so the cost is higher.
Long run, the disruptions for both British and French spheres can be weathered. the question becomes whether the ripples of these disruptions cause massive unforeseen butterfly effects.