So after struggling with counters for a long time, I had an idea, which was an easy way out, but for some reason it did come up earlier. Now I can’t see the game and the designs without this Counter. And it proved to be working during the playtest.
So, the idea is that the counter is a set of cards, three for Mr Dream and three for Ms Reality. Each time when during the round Reality wins, Mr Dream and Ms Reality players put a card of Reality face down on the table. Same with Dream. When 3 Rounds are played, they mix the cards and reveal them to all the players. And only after the cards on the Game Counter are flipped.
Not only the Round Score Counter was changed into a card format, but the Game Score Counter too. It made the whole process more comfortable, as everything was now adjustable to different, but proportional sizes.