How do I set a custom ringtone on the Pixel 2?

It's quite possible that the song name is embedded in the file and that the OS was exposing that rather than the filename. Unfortunately you've confused the matter by moving it somewhere else, and relying on the Windows directory structure.
I'd suggest removing the file completely, rebooting the phone to make sure that name (The Theme...) is no longer shown, then rename the file to something random (that has nothing to do with Peter Gunn) and put it into the folder that B. Diddy recommended. If it shows up in the Ringtones list with the name that we're expecting (The Theme...) then you'll be confident of the system behavior.
 
It sounds like you did move it to the correct directory, though. The full pathname for /Ringtones on the phone would be /storage/emulated/0/Ringtones or /storage/sdcard0/Ringtones, but from the PC, /Internal Shared Storage/Ringtones sounds like the same thing.

I've seen this before on older phones, but can't say that I've checked on my Pixel 2 XL -- if you add a custom ringtone and select it for use, then delete that custom ringtone file from the /Ringtones directory, the tone can still remain in the selection screen. It may be that the system saves it (at least temporarily) to its own system directory along with other preinstalled ringtones. So if you add the file again to /Ringtones, it'll show up twice in the selection screen.
 
Thank you both for sticking with me on this. The hex editor reveals that the original file name ("Theme Song to Peter Gunn") is embedded in the file. Strange but explains a lot. Thanks again for your assistance and perseverance.

Bob