Hey Gilles,
No worries about testing with another Mac.
Since another Mac isn’t available, we'd like to try to isolate the issue a bit. Please follow the steps below:
1. Disconnect all eternal devices.
2. Forget all Bluetooth devices except for the Magic Keyboard: Connect a Bluetooth device with your Mac - Apple Support
3. Connect your Magic Keyboard with Touch ID to your Mac using your USB to Lightning cable.
4. Restart your Mac with only the Magic Keyboard with Touch ID connected (and a mouse/trackpad if needed). When you get to the sign-in screen, make sure the keyboard is working as expected before continuing.
5. Leave your Mac wake, connected to the internet and do not use it at all for a minimum of 30 minutes.
Once the 30 minutes have been completed, restart your Mac and see if Touch ID is working.
If you still need to restart your Mac again for Touch ID to work, please contact Apple Support directly: Get Support
All the best.