After the Napoleon's retreat from Russia, he had been severly weakened. Even if the Swedes allied Napoleon, he would still have a huge problem with the whole of Europe (except Denmark and French puppet states) turning against him.
Besides, as was mentionned, the Swedes would have trouble crossing the Baltic as the Royal Navy's in the way. The Swedes still have their fleet of course, but the Royal Navy was clearly superior.
A more interisting question would be what if Bernadotte had invaded Finland as Napoleon had asked him to when he invaded Russia? The Russians would have to send some troops there because the Swedes could threaten St.Petersburg. This might not guarantee Napoleon's victory in Russia, but it could increase his chances after the Russian Campaign.