Update:
I went back to the same fast-food restaurant today for lunch...
Instead of saying AMEX, I just said credit as I assumed it didn't matter what the card type was. Sure enough, I was right, and the transaction went through on my card without issue. No need to use the Visa this time! And now I know for future if they ever tell me otherwise, but I also hope that the employees won't be reprimanded by their boss for unknowingly letting it happen.