However, I think I'm mistaken - transactions can't access the hash of the block they're in, since it's only computed after the transactions run. It'd still be trivial for a miner to game this, though, by only including the transaction if it causes them to win.
Theoretically may be. But I don't think it is really possible nor trivial in practice. There is not only one miner. And while the one tries to find a hash with and without his transaction other miners will already submitting new blocks. And btw. it would not be worth the amount he can win ;-).