Who succeeds Charles XIII as King of Sweden without Jean Bernadotte?
Depends a lot on the ifs ands or buts. It's not impossible that another french marshal could be found, a lot of the Swedes wanted a "warrior king" and the French had the best army in Europe at the time. It'll take someone with a bith of a knack for charm and bluff like Bernadotte though.
If not a french marshal, then the natural replacement for Crown Prince Charles August is his older brother, Frederick Christian. He candidated for it otl until Frederick VI opposed him, since the King believed he might be elected himself. Frederick VI's candidacy was never really considered a viable option in Sweden though (afaik) and if this becomes clear Frederick Christian might give it a go anyway. If he dies as per otl in 1814, then
his natural successor is his eldest son Christian August, otl claimant to Schleswig-Holstein during the three-years war.
If neither a french marshal nor the Augustenburgs, then perhaps Charles XIII's cousin Peter, in 1810 regent of the Duchy of Oldenburg for his mentally ill brother. The problem might be the continental Holstein-Gottorp's close ties to Russia, against whom Sweden is intensily revanschist at the time.
Finally, if truly no heir can be found, and keeping the monarchy is a must, they may just have to bring back Gustav IV's son, the otl Duke of Wasa. This would probably be the last of the last ditches, since basically none of the men who had deposed his father would be in favour of it, but if enough time passes and Sweden falls into a succession crisis he might have a chance.
Other than these I can't really think of any. You could perhaps go all the way back to the Hesse's and find a prince there, but at that point you may as well just throw a dart at a map and pick some prince from wherever you hit.