That did not help me much to understand what are you trying to achieve. Having shared account for a training session is understandable, but I don't get the rest 🙂
Anyway, you could create a simple script that will do a Start-Process powershell.exe -credential (get-credential) parameter and start your second script that will write the info to the file. If the start-process fails due to bad authentication, you can logoff the shared account.
Make sure to enable this GPO option otherwise your users will not get the interactive Powershell window.