@CallMeX Welcome to the forum!
To be clear, are you asking how to allow the biometric unlock to be used when you first unlock the browser extension after starting (or restarting) the Chrome browser? Can you confirm that the biometric unlock of the browser extension works as expected after you have first unlocked it the first time using your master password?
If your question is just related to the unlock behavior after restarting the browser, then open the browser extension options, and check whether the option “Ask for biometrics on launch” is enabled or disabled. Frankly, the documentation for this setting is a bit confusing, so I would suggest trying to toggle this option (disable it if it was enabled, or enable it if it was disabled) and checking of that made a difference.