Shiny Server Pro does not support SSO out of the box (as opposed to it's successor RStudio Connect). What you can use is proxied authentication, which requires you to forward the user name in an HTTP header. It sounds like your reverse proxy only authenticates the user via SSO but does not inform Shiny Server Pro about that.
If you don't understand these terms, it is best if you find the IT department within your organisation which is responsible for the reverse proxy. Show them the documentation linked above and ask them if they can implement something like this within the reverse proxy.