I'm not ready to suggest the right rules, but the scale of the battles was rather small.
Ermak's detachment consisted of 540 Volga Cossacks. In addition, the Cossacks usually had one or more "chur" - the so-called adolescent teenager, who should have earned the right to be accepted into the Cossacks. The structure of the detachment Ermak included about 50 people from the military squads of merchants Stroganov, among whom were prisoners of the Livonian War (it is possible - Europeans). The total number of Cossack detachment is about 1000 people.
Later, Ermak received reinforcements from the tsar, a detachment of Prince Semyon Bolkhovsky. The squad consisted of 300 streltsy from Kazan and other nearby cities.
The Cossacks had in their arms a pischal (an analog of the European arquebus), small cannons. The main cold weapon used by the Cossacks is a spear. All the Cossacks had a good protective armor, which protected them in battle against the arrows of the Siberians. The Cossack boat carried about 20 people. Widespread in the Cossacks way of fighting - the best shooters are firing, the rest are charging and giving the guns to the shooters, which allows for a fairly intense fire.
The army of Khan Kuchum, according to recollections, could reach 10,000 people (which is clearly exaggerated). In any case, it was much larger than Ermak's detachment. Its core was the Nogai Guards (about 400 people) and the Tatars. It was a beautiful steppe cavalry. But there was not much of it, since the son of Khan Kuchum - Aley, left with a large part of the troops to fight with the Russians for the Ural Mountains. He returned only the next year. Local tribes Khanty and Mansi were auxiliary pedestrian detachments. The forced Siberian peoples did not want to fight for the interests of the Tatars and therefore had low fighting qualities.
The main weapon of the Siberians is a good bow, similar to the Mongolian one. Warriors Kuchum preferred to distance combat. The cavalry could attack with cold steel.
Firearms were familiar to Tatars and they were not afraid of him. The Siberians often met with this weapon in raids to the east of the Moscow principality. Khan Kuchum had even two own cannon, but there were no gunners for their use.
I hope this helps you with the drafting of army lists. If you have any questions - ask.
