
I can’t tell you how many times I’ve had to go back and pick up my Gloomhaven character because it fell over. The tiniest slope can cause it. I can imagine several ways of fixing it, like allowing some tokens to be designated as upright-only, or perhaps just moving the center of gravity to the base.

Have you tried setting the game physics (top bar ‘Options’>’Physics’) to ‘Locked’? This option prevents most of physics interactions for the objects that are stationary.
PS: IMO, this option should be set by default. Very rare board games require physics simulation for stationary objects.

I have! It mostly happens when you move your character and set it down so it overlaps part of another token, or if the map tiles aren’t lined up quite right and have a slant.