Aren't "Sami" and "Suomi" the same word? Aren't Finns, to oversimplify, those Sami who took up agriculture?
No.
The languages are distantly related, being Uralic laguages. They separated almost 5000 years ago. The Saami language only displaced the previous/original laguage of the Saami about 1500 years ago. Previous to that, they most likly spoke a laguage isolate. The Finns seem to have a continuity with the Finnish language.
Genetics indicate that the Saami went through a period of genetic isolation that may have exceeded 10 000 years at some point.
As for the OP, the Saami suffered far less in the Black Death than the Scandinavians. It may be the point in time where the demographics best favored the Saami.
However, I would cosider something more modern. The Saami are disadvantaged by a very low popolation density. Basically the old nomad/agricuturalist problem still persisting in an area of Europe where the climate prevents the agriculturalists from getting a decisive advantge.
Borders here have been fluent and rather theroretical up to not quite modern times, and its not been unknown for three nations to try to tax the inhabitants.
I would think the best option is to have a greater power set up some kind of buffer or vassal state initially. Russia or Britiain, to counter the other, or restrict Sweden or Norway at something. Maybe a different Napoleonic wars.
Then have the Saami expand more into the Scandinavian ecological niche of potato farming, as well as expand the fishing. That sustains a higher population density. Whaling followed by extraction of mineral resources fuels modernization.