Bold Memberships might be a good option here. You can set it up to have multiple membership tiers and you can give access to specific pages for members only. So a possibility would be only allowing members to have access to the game page. Then to track, I would think some sort of auto-tagging would need to be in place. Hope that helps!
Yeah happy to help! You can tag a customer when they purchase a specific product. So one possibility may be to 'sell' the game as a free product. So when they buy it they can access the game and you can tag them as someone who played it. You could also tag a customer when they purchase a membership.
Not 100% sure how you would set it up, but just throwing some ideas your way.