A malicious set of miners that decides to give itself a higher block reward will never have their blocks accepted by the "good" nodes. It's quite simply not possible. The blocks will be invalid and therefore rejected outright. It doesn't matter how much hashing power is behind them.

> If they're bored, they can devote some of their hash power to double spend attack the "good" chain. Which chain will come out on top?

That's a different attack entirely.