Russia may not be the most keen on the idea however, having a large power in control of the straits may push Russia to joining Prussia's side. Unless Sweden and Russia can come to an agreement of lower tariffs. It was control of the major Baltic River trade routes during the 17th century that produced the biggest amount of income for Sweden. Would a union be able to bring Sweden back to economical height?
Well, Sweden is just about the richest EU member state today - going by IMF estimates for GDP per capita, it ranks fourth. It may not have colonies in the Baltic or in northern Germany, but it's doing fine, thank you very much.
In a Scandinavist TL, it'd be even richer, because of the oil. Norway was the poorest Scandinavian country before it discovered oil, and when it was exploring, it asked Sweden for help and offered it a substantial share of the proceeds, since there was more technical expertise in Sweden. Sweden refused, and Norway's reaped a windfall since.
The one hitch is that Stockholm is a poor choice for a pan-Scandinavian capital, especially if Finland is still under Russian control. I can see Denmark and Sweden compromise on Oslo as a national capital, or alternatively choose a smaller city at the border between two countries, probably Malmö. They might even put each branch of government in a separate city, like South Africa.