I use anonymous posting for all visitors. Only the admins and moderators need to log in. As far as bbPress is concerned, an anonymous user is identified solely by a “name” and an email address. Once that combination is used, any time they visit again, they will be identified. If they change it to anything else, they are not recognized. It actually works quite well. However I have found that users who do a lot of social media get so confused about this and try to log in all the time, locking them out of the site after so many tries. I finally had to put up a pop up on the forum page that reminds them that they may post anonymously, only a name and email is required in the Create New Topic or Make New Reply.
However, this won’t work for a hidden forum that is accessible to only logged in users. I have several forums besides the public anonymous forum and those forums are for logged in users only. One is for the moderators and another is for users who have special needs for privacy and I make them a username and and password, which I email them.
You can set this under Settings>Forum. Look for this line and check it to set your forum to allow anonymous posting:
Anonymous posting: Allow guest users without accounts to create topics and replies
Make sure you have Akismet plugin installed and activated. I also run WordFence for extra coverage. Between the two, I never have any problems with anon posting. I’ve been using this like this for 7 years.
If you want your normally logged in users to be able to post anonymously to a public forum, you could have them use a different name and email combination, making sure the anonymous posting setting is set under Forum Settings. I have myself set as keymaster with one combination and I use another name and email so I can post anonymously for testing purposes.