I am offering an improbably challenge. The idea is to give out an extremely unlikely (but not impossible) scenario, and then try to find the most realistic way to get it to work. So, here goes.
The challenge is to make Islam the dominant religion in Scandinavia by 1300, without causing a Years of Rice and Salt-style uber-plague. Also, the religion must be accepted by the majority of the Scandinavian population, not just by the upper classes.
Ready... go!