You just need to exclude the base forum path: /forums* :

Here are information how to exclude it:

https://www.webnots.com/exclude-pages-and-directories-from-caching-in-w3-total-cache/

We just realized we had a similar wpForo login/caching issue. I guess since wpForo isn’t built to “tell” W3TC when a page/post/comment is updated, W3TC has no way of knowing the same, so it doesn’t update the front end.

This isn't a dig at W3TC or wpForo, just an "aha!" moment that just because these two plugins are very tightly integrated with Wordpress doesn't mean that Wordpress tells one about the other's actions.

Adding (for us) /community* to W3TC's Never cache the following pages: list seems to have solved this for us. 🙂