Use server logs instead of client side libraries
One way to do that is to setup this package:
Do you receive server logs from your CDN? If not, maybe you can serve something from your own servers to have some server logs under your control.
This is exactly the approach I was thinking of! It just sounds like a lot of work (and some costs) to set up and maintain a server for this purpose, and then more work to produce useful analytics out of server logs with some measure of "unique" visitors. I was thinking maybe somebody else has already created a product that works like this.