What I'm wondering is can I simply transfer the BTC from my Electrum into my Ledger Nano S (instead of transferring to a new Electrum address) and then install Electron Cash (on a different machine, although, post-successful-transfer, I doubt it matters, but better be safe than sorry) and use the old Electrum BTC address to claim BCH from?
Afaik, yes. That's mostly the way I did it too.
Sending the BTC to a new address first is a precaution to avoid losing it by any chance. The second step is importing the seed or private key into the BCH-enabled wallet (such as Electron Cash) and transfer the BCH to the newly generated BCH wallet. Just remember not to use old addresses anymore (from which you claimed BCH).