YourStepDad said:
If there were a way to flesh them out though, such as with unique Finnish warcries or possibly uniforms or armor, that would always be nice. As far as I know though equipment wise they indeed were generic reiters.
I actually tried researching what the hakkappeliita were equipped with but found no real sources on this whatsoever.
There are some suggestions/evidence that the Finnish cavalry was equipped differently from the Swedish, that they could have breastplates and helmets as depicted in these photos.
http://hem.bredband.net/petterssonpether/1/hackapells.jpg
https://upload.wikimedia.org/wikipedia/commons/7/7b/Hakkapeliitta_2006_b.jpg
Perhaps a completely new unit wouldn't be needed, but maybe some new equipment for the Swedish reiters as well as a new horse (they used Finnish horses that were smaller and more durable than German ones, for example).