Thanks, this worked! I am back to being able to open random links in a private browser rather than in my primary browser with this registry file applied and changing HTTP/HTTPS to use it (which becomes available immediately after applying this registry file). The same general trick could likely be used to make it so a specific browser profile is used to handle links rather than the default profile.
I did make one minor fix to it which I recommend applying to the original gist:
https://gist.github.com/winhelponline/2f39276cdb1872304f72ecd843476d71/#file-firefox_private-reg-L12
That should end with private_browsing.exe,0" instead of firefox2.exe,0"