You can not update the style of the browser picker at the moment.
There isn't (currently) a way to modify it.
This is one of the browser special permissions dialogs (See https://permission.site/ for more). I believe it is vital that these dialogs cannot be forged or be misleading to maintain trust with the users- although I wasn't able to find anything in the spec calling this out.
I first recall this being discussed around the Permissions API- however I only could find this note:
NOTE
This is intentionally vague about the details of the permission UI and how the user agent infers user intent. User agents should be able to explore lots of UI within this framework.