"Access to the registry key 'Global' is denied" when accessing performance counters

I found the solution. Whereas various links around the Internet tell you to use the identity IIS APPPOOL\DefaultAppPool, that is NOT the full story. That will only work if you are in fact assigning your applications to an app pool named "DefaultAppPool".

This link at ServerFault has the answer. You must use the name of your app pool. In my case, the app pool's name was "staging", so the correct user was IIS APPPOOL\staging. Putting this user in the Performance Monitor Users group fixed my issue.