I think that on balance I would agree with Rogerc. Unfortunately I think that most battles that are easy to find information on are either larger than the 6 to 10 units you want. There are some in all areas but information can be very difficult to find.
Historically the Russian armies are big and usually they need to outnumber the Swedes by quite a bit. I am afraid I am not a fan of Beneath the Lily Banner but even with these rules I believe that you would need to have more of these to make a game of it.
Both the Danes and Saxons have smaller armies, although usually outnumbering the Swedes again. But not by much and the armies are smaller. While 6 to 10 units would be a relative small part of a Russian army it would be a substantial part of a Danish/Saxon army. The full Danish/Saxon army would be something like 25 to 30 units as a rough guide. So while I think you might struggle to find historical battles with 6 to 10 units in total it will be easier to find battles were 6 to 10 units are the right wing (or some other section) of a full battle. Similarly scaling down battles will be easier.
I think the Saxons would edge things because of the reasons Rogerc suggest (by the way his missed off the prospect of Polish allies for the Saxons - there are some lovely examples of Poles on his blog). I would add they were also the main opponent for the Swedes up to 1706 and were active in most of the war. The Danish participation was more sporadic and also mainly at a time when the Swedes were very much on their last legs. But they too would I think suit your purposes if you fancy doing them.
On the skirmishing idea then I would guess any of the armies would be fine with whatever 'main' army you went with.