I can't imagine anyone being willing to download the 185 GB Bitcoin database just to play a game. Perhaps 185 GB will seem tiny in the future, but for now i don't think your idea is feasible.
If the gamer will receive some money (satoshi) for participating in distributed mining - why not?
Mining Bitcoin with a CPU is no longer economically feasible, currently earning less than $0.002 per year. Who would pay the gamer to mine, and why?