"Double-click Touch ID to pay" doesn't work on MacBook

At some point during the macOS Sonoma release, my Apple Pay dialog in Safari began saying "Double-click Touch ID to pay" instead of the usual process of simply touching the Touch ID sensor to approve the payment. Trying to do this would immediately lock and unlock the screen instead of process the payment. If I'm lucky, I might be prompted for my login password instead. How can I make payments with Apple Pay in Safari on my MacBook running Sonoma?