Keep BW vault unlocked across browser refreshes

You can change the vault timeout to ‘never’ and accomplish this - you can also use the shortcut CTRL+SHIFT+N to lock the vault manually.

More on vault timeout options here: Vault Timeout Options | Bitwarden Help & Support

There is a feature request here to have a ‘central’ desktop session more or less: Login to browser extensions when logging into desktop app and vice versa