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Shameless plug: My project, Cactus Comments[1] does something similar, although for commenting, instead of support chat.

We ship a web-embeddable Matrix client. The users' browser connects directly with a Matrix server of their choice (or a default one).

This is very close to what I had found a need for today - now if only I had a (self-hosted)-wordpress-user-accounts bridge for the login -

and auto-prune to delete all messages after X number of hours - a new project could happen quickly with this.