BuddyPress.org

I hate repeating myself but the reason I mention the renaming is BuddyPress uses the “forums” slug by default, so if you installed standalone bbPress at /forums/, there will be a slug conflict.

Did you even install standalone bbPress?

On a properly-integrated bbPress and WordPress install, you can login at standalone bbPress and you will be logged in at WordPress and vice versa. I’ve done this on multiple occasions and it works.