Ive been using BW on Safari ( > 8TB shh) without problems. I enabled 2FA with Google Authenticator.
It worked for 3+ major macOS releases ( Mojave, Catalina, BigSur ). After updating to Monterrey, Ive been aLSO using BW with Safari till today.
Started Safari, clicked on BW extension, on login prompt entered login ( email ) + master pass; enter, and this:
Same action on vault
On other browsers ( Vivaldi, Brave, pure Chrome ) visiting vault and entrering my login credentials brings me to enter 2FA code screen.