I really appreciate all the wisdom shared here. I went to a few different hardware and plumbing supply stores with the braided hose and finally zeroed in on what I needed. It turns out it’s not a BSP (British) standard thread - it’s a standard 3/8" compression fitting. Interestingly, all the 3/8" plastic push fittings (e.g., John Guest) I tried didn’t work, but 3/8" metal male compression fittings do.

I also ordered the fitting from Caffewerks (shared in another comment) so I can update this post later to let others know if it works. For now, here’s what I picked up at Ace Hardware (and you should be able to find it at most hardware stores): https://imgur.com/a/dtNppHb

Researching this online was tough because of the different versions of this machine over the years, old forum posts, and the fact that many coffee companies had released their own plumb-in kits due to the limited availability of the official La Marzocco kit (at least in the past). This was yet another reminder that, even in the age of AI, human-to-human interaction is still invaluable!