To be fair:
You shouldn't rely on the randomness of the hash if people are betting on it. Especially not after the hashing.
If you have a miner who stands to make more on the bet than they would on the block subsidy, the economics can favor them getting a hash low enough to make a block, but then withholding it, looking for a hash low enough to make the block *AND* having a final digit that will win them the bet as well. They risk losing the block subsidy if another miner finds a block first, but they can also instantly release the block they found the first instant they get a whiff of the other miner's block, and then they have a nearly-equal chance of getting the block subsidy anyhow. Meanwhile, they go on looking to win the bet.
So, according to you, if this game goes big, miners may not publish valid blocks and thereby disrupt bitcoin block generation altogether?
That is if they have sufficient resources to even generate a block. They would probably need a significant hash rate to be able to generate blocks at a reasonable frequency. The miner can always mine the block and start mining on it before publishing it. The miner have a chance to publish two blocks at once and no one else could've solved on it. This theoretically can give them a higher chance of getting the subsequent blocks.
The method mentioned above is quite effective for the attacker and would have a significant chance of succeeding if you have a good amount of hashpower. However, if you do not have a good portion of the network's hash power, your probability of succeeding will be close to insignificant. If someone mined the block when you are withholding it, unless you have a high amount of hash power, more often than not, the block you mined would be orphaned. That obviously depends on which block the miners decided to work on. There is a chance for them to lose both the block reward and the bet altogether.
Miners would have to continue to publish the blocks since the game uses it to determine the results. Bitcoin's block would at the very most be delayed.