> Are you saying this would be running on the machine that runs the relay?

It can, doesn't have to. If it's on the same machine, use the local ip (127.0.0.1):7777.

> So that Tor traffic can find the relay?

Tor traffic finds the Tor hidden service. The hidden service forwards the traffic to your relay

> Does the client UI then need to open websockets to the Tor URL, rather than the clearweb URL?

Not 100% sure on this. I think the Tor browser takes care of this for you. I'm able to get it working with Nostrudel on the Tor browser.