Charles August's alt-history potential has always intrigued me, I'm actually a bit surprised that he isn't used more often. I think this stems from a few places, Bernadotte was such a curve ball in otl regarding Sweden and the Napoleonic wars that it becomes difficult to predict what would've happened without him. August also has both short- and long-term possibilities, but I think it can be difficult to gauge how these would affect eachother. Also I think he has kind of a reputation of being sort of a loser, especially compared to Charles XIV, and this may turn people off of using him.
I did start a thread a while ago about Bernadotte not becoming King of Sweden and it’s effect on the Napoleonic war and the Russian campaign specifically. There wasn’t that much discussion, but there was a good case made that it wouldn’t drastically change Russia’s situation. Ultimately if Sweden sided with France in the invasion, it would likely be ruinous for the already weakened country and it may end up under Russian occupation in one way or the other.
But it’s not at all certain that Charles August would take Sweden in that direction, sure there was a large portion of Swedish leadership who were in favor of an attempted reconquest of Finland, but certainly not all, and even some of the hawks realized that even if war was desirable it wasn't feasible at the time. My personal belief is that Charles August entertained the hawks while he still wasn't secure in his position, but that he didn't intend to make good on their hopes. In short I don't think Sweden under Charles August joins France in a war against Russia.
But it does raise a lot of questions about what direction he would take Sweden, he was in favor of Scandinavian unity - initially he supported Frederick VI's candidacy to the Swedish throne, but once that clearly wasn't on the table he also seems to have had ambitions for taking Norway from Denmark. However, Frederick VI planned to start negotiations for Charles August to be engaged to his daughter Caroline shortly before his death, which might not mean war with Denmark after all.
Disregardning how the Napoleonic wars end for Sweden and Denmark-Norway, Denmark and Sweden likely become much closer dynastically since the Augustenburgs are likely to inherit in Denmark sooner or later, especially since the reason for the enemity between them and the main Oldenburg line is gone with Charles August still alive. If the stars align just right then pan-scandinavian sentiment may coincide with a personal union between Denmark(-Norway) and Sweden(-Norway).
EDIT: Another short-term consequence of Charles August dying is of course Axel von Fersen not being lynched, strengthening the Gustavians, again I'm unsure how much this would affect, Fersens time as a prime political mover was behind him anyway, but Charles August could align himself with them to strengthen his personal rule. I don't see Gustav Gustavsson being adopted as an heir as a possibility, but it's not impossible that he could be rehabilitated in some way down the line.