I disagree that one should expect Bitwarden to respond to double-clicks differently from the browser in which it resides.
If you wish for Bitwarden to implement some idiosyncratic behavior, please open a feature request (but be prepared to present a cogent case for why Bitwarden should devote development resources to such an effort).
Alternatively, if you don’t wish to use the “Copy Password” buttons, you could select the password via click-and-drag, or via triple-click.