How to access fingerprint to clamshell when using an external monitor on Mac?

You don't.

There are some Macs that have the right security chip so that they can work with the Apple Bluetooth keyboard with "Touch ID" finger print reader35220145.

Magic Keyboard with Touch ID for Mac models with Apple silicon - US English - Apple

and

Magic Keyboard with Touch ID and Numeric Keypad for Mac models with Apple silicon - US English - White Keys - Apple

NOTE: There are non-US models. I just posted the first 2 I found.

Compatibility

Mac Models

• • MacBook Air (13-inch, M3, 2024)
  • MacBook Air (15-inch, M3, 2024)
  • MacBook Air (15-inch, M2, 2023)
  • MacBook Air (13-inch, M2, 2022)
  • MacBook Air (M1, 2020)
  • MacBook Pro (13‑inch, M2, 2022)
  • MacBook Pro (13-inch, M1, 2020)
  • MacBook Pro (14‑inch, 2023)
  • MacBook Pro (14‑inch, 2021)
  • MacBook Pro (16‑inch, 2023)
  • MacBook Pro (16-inch, 2021)
  • iMac (24-inch, M3, 2023)
  • iMac (24-inch, M1, 2021)
  • Mac Studio (2023)
  • Mac Studio (2022)
  • Mac mini (2023)
  • Mac mini (M1, 2020)
  • Mac Pro (2023)

And I think you can use Yubikey USB devices as a substitute, but I am TOTALLY unsure, as I have never actually used one of these devices

About Security Keys for Apple ID - Apple Support

Lets see, there are some password/finger print prompts that can also go to your Apple Watch if you have it configured to unlock your Mac.

Or you can do what I do, which is use your MacBook (Pro, Air, etc...) as another monitor.