Their site indicates that the free tier requires adding a JavaScript beacon (which is what I’m trying to avoid).

Is that what you’re using? Or did I misread their offering (which seemed to clearly indicate a paid plan was required for the non-JavaScript analytics)?

See at the bottom how it says “sign up for a paid plan” in the section that says “there is no code to add”:

https://www.cloudflare.com/web-analytics/

Edit: Oh, you're the OP! Just checked out your site and you are indeed using their JavaScript beacon. I know I'm being unreasonable, but in my case, I'm going for no JavaScript, and no trackers (so some sort of request logging on the server or proxy side is likely my only option). It's my attempt to "be the change you want to see in the world".