Run app as administrator on logon of non-administrative user accounts

Question

We need a program / application to open with local administrative permissions (Process Monitor, so it's non-negotiable) on logon of any given (including non-administrative) user account.

Task Scheduler seems to be the way to go with this but I'm having major problems bypassing the UAC prompt:

1. Option When running this task, use the following user account seems to determine which user account / profile the process will run under so:
   1a. Using Administrator, SYSTEM, etc doesn't work as required.
   1b. Using Users does work but...
2. Logged on as a non-administrative user account, option Run with the highest privileges does prompt for credentials but fails due to insufficient permissions to save the task.
3. Logged on as an administrative user account, option Run with the highest privileges doesn't prompt for credentials and, as such, the app opens with a UAC prompt on logon of a non-administrative user account.

  Not duplicates of:

1. Force a program to run *without* administrator privileges or UAC? or How to launch a GUI Windows application with administrator rights without asking for the pasword from the user? because the app must actually run with administrative permissions.
2. How can I run ALL my Apps “as administrator” by default in Windows 7? or Run applications as administrator by default in Windows 10 because only this app must be run with administrative permissions.