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I'm having trouble wrapping my head around this. Once you use your private key to complete a transaction on the BCH blockchain, won't that reveal the private key publicly and allow others to spend the corresponding unspent bitcoin on the BTC blockchain? Sorry if I'm overlooking something obvious, but it was my understanding that once someone used their private key it should never be used again...

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Not at all. You never reveal private keys when transacting. You will reveal addresses, however. The only BTC addresses that have BCH value are those that existed at the fork. So before using them in the BCH blockchain, it would be prudent to first spend them in the BTC blockchain. Doing this is a double spend in some sense, but they happen in different blockchains, so they don't affect each other.

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