If the Free French forces are so significant that GB can think about invading then Vichy is dead and will be killed off by the Germans as its to risky to have any formed French forces that could swap sides attacking from the south (ie proto dragoon without the sea lift) to support an GB/FF invasion from the north.
Even if botched and half-assed as OTL Dieppe Raid? I am toying about a less popular but belligerant Vichy where FF's landing failure bring Laval to power again.