Strategy
Mouse
If the shaman is not competent, the platform must remain balanced. Try to stay on the side with the least amount of mice, because even if the platform tips, you will have the highest chance of survival if you head toward the center. If you're caught on the heavier side when this happens, you might be able to wall jump, but this is very difficult against a falling surface. Switch sides if yours starts to descend, and once you have the cheese, make a mad dash for the hole. Alternatively, stay in the middle until the shaman stabilizes the platform.
Shaman
As soon as possible, stabilize the seesaw with a red-anchored plank in the middle, just below.
But if the platform falls over before you can manage this, try to put a plank on the middle pillar and go on it so that you can bridge the surviving mice to the cheese and back.
Updates
V0.148 - The pillar of wood is now transparent.