There are a number of ways Charlie can play this hidden card.
1) He can start mining when no one else can and win every round. If he adopts this strategy, he can win at most 10 times in a row. Since he has to doubling his personal difficulty each time, eventually he will stop. If ENC prices hold during this period, they will continue to hold. If they move some, they will hold at the final price once Charlie runs out of arbitrage room.
Hmm I'm still confused about how difficulty scales in your system. Why can Charlie win at most 10 times in a row? And what do you mean by doubling his "personal" difficulty?
In your proposal it says that difficulty retargets to current+1 if a proof-of-work+1 transaction wins and goes down to current-1 if no proof-of-work is submitted, so wouldn't it be better for Charlie to pump up difficulty to monopolize and then just mine proof-of-work+1 one block then stop the following block to bring difficulty down, rinse and repeat?