Apparently I had another issue -- my name has a non-standard character that is correctly recognized on the Eurostar website but not on the Uber app, which meant that when linking the club # it wouldn't accept as the names didn't match.
It's all solved now, and I can confirm that Uber bookings do give Club Eurostar points (which means you can stack them up with the 10% back in Uber Credits, which is good news)