Make sure you have no login, register widget on the WP side and use a plugin for login,regsiter that re-directs users back to the page they logged in from. There are several that do that.
I even believe there is a bbPress login widget
I presume you have on the WP side comments and such off for users (only read blog)
I need to be able to first approve of their membership, so it can’t be automatic, and then I have to add them as users. Is there a widget/plugin that allows me to add them as users manually? When I try and do it manually on the WP admin dashboard, it also forces me to make them a user on the blog, which I don’t want. I want them to only access the forum.
https://wordpress.org/plugins/new-user-approve/
There might be other plugins also just do a search
On the WP side you can always just keep them at -No role for this site- or just subscriber, which just means they can read, but not write posts on the WP side.
When I try and do it manually on the WP admin dashboard, it also forces me to make them a user on the blog, which I don’t want. I want them to only access the forum.
bbPress users ARE WordPress users so as @ronthai points out just make the users WordPress role ‘Subscriber’ to which point you may as well just use the built in users panel in your WordPress admin.