@cadenpalmtree I tried it again and this time it worked. I assume they've updated either iCloud for Windows, the Chrome extension or both and that has resolved the issue.
Sidenote: To get the iCloud Passwords App inside of iCloud for Windows to sync my passwords I had to log out and log in again.
Thanks for pointing out that it worked for you!
As I said earlier, I switched over to Microsoft Authenticator on all my devices but it's good to know that I have iCloud available as a fallback solution.