Is there a reason why you don’t just directly unlock to the browser extension and launch the URL from there? And you can save additional steps by using the Ctrl+Shift+L shortcut to autofill the login credentials.
The workflow should look like this:
Click browser extension icon (or type Ctrl+Shift+Y) to open the browser extension to its unlock page, and enter the password or PIN to unlock.
Search for a login entry in the browser extension, and click the “Launch” icon to launch the site.
Type Ctrl+Shift+L to autofill the login credentials, and hit Enter key to complete login process.
Perhaps the gist of your request is that you would like for the unlocking of one Bitwarden client (e.g., the web vault) to also unlock all other clients (e.g., the browser extensions). If so, you may want to vote for the existing Feature Request linked below: