Currently, there is the "wpForo Private Messages" add-on:
I don't know familiar with live chat plugins, can suggest nothing. You can install and check, I think they should work fine.
I would like a "chat box", too. Something like the shoutbox of MyBB:
And I'm sorry to say so, but: Most chat plugins from WordPress.org are not particularly suited for chatting among members and either look old and are no longer updated, or have a subscription model for several hundred Euro a year. I also like the proposal from Percysgrowroom pretty well. :D
Of course, if that does not work or is not planned by your side, that's just how it is. But the proposal is already there.
MyBB Shoutbox (old design):