Can I simply download the BCH wallet and replace the wallet.dat with the one in my BTC wallet?
Yes... If you have been using Bitcoin Core, then the wallet.dat from that will be compatible with the BCH "Core" wallet.You'll need to download BitcoinABC or whatever the "core" BCH wallet is these days (given the whole BCHA and BCHN bunfight as to which was the "real" Bitcoin Cash
)... then put a copy of your Bitcoin Core wallet.dat into the datadir for the BCH wallet app. Then open the BCH app and let it fully sync.