Finally found what I needed!

I tried a large number of the chat room plugins. I found a few that worked… mostly. But when I tried to apply them to my specific needs, they just weren’t flexible enough.
Wise Chat not only worked properly right out of the box, but it was also very flexible. The coding is not irrationally obscure either. I was able to dig in and customize it in a logical and controlled manner.

Well done to the developers!

PS. Let me know if you create one using websockets like you had mentioned.