Some thoughts from a Finn.
King of Sweden, Gustav IV Adolf, was kinda bonkers and thought that napoleon was the Anti-Christ and saw himself as the Anointed of the God who was going to rid the world of this monster. So I don't see it very possible that he'd ally himself with The Beast of the Apocalypse.
Losing Finland in 1809 (due to the war of 1808) was a catastrophe for Sweden, but stroke of luck for Finland. For hundreds of years Finland had been a shunned backwater for Sweden, but since the Russians needed to appease the just conquered Finns so they gave the country a lot of autonomy and Finland was an autonomous grand duchy in Russia. Although this autonomy was later on attacked it was still better than with the Swedes.
a hundred years later Finns got independence which I fail to see happening with the Swedes.
If Finland would've stayed with Sweden, in 1939 Soviet Union would've still demanded areas for the safety of Leningrad and if Swedes had agreed I think the whole country would've been annexed the way Baltic countries were and if Winter War had broken out, the miracle of the Winter War defense would've never happened (even if with Sweden Finns would've had more resources) as it was a very much a national thing and under Swedes there wouldn't have been as unified a people as with independent Finland. So whatever happened in Winter War Finland would've ended under Soviet yoke in 1939-1940.
As Hitler apparently got the idea of attacking Soviet Union from the abysmally poor performance of Red Army during the Winter War (or at least that showed him that the Soviets were in dire straits) Barbarossa might've never happened, had Soviets wrestled Finland from Swedes.
If it had it is rather probable that Germans would've invaded also through Finland, which would've let the country in much worse condition than it now was after Continuation War ended in 1944.
Perhaps Finland would've finally gotten the independence in early 90's along with the Baltic countries when Soviet Union crumbled. There would be plenty of tension in the borders today and Finland most likely would've joined NATO along with the Baltic states which would also increase tensions.