Can I RDP through mstsc and run a script on that remote desktop?

No. MSTSC is a terminal client. You should really get enter-pssession working if you want to run scripts on remote machines. It will make your life much easier.

It may be as simple as you just need to allow Powershell Remoting by running a single command on each machine:

Enable-PSRemoting –force

You can create a scheduled task which will be triggered by logging into the server using a special account within in domain and by the trigger a system will run a prepared script which will be located on each server independently.

Steps: You can create a scheduled task that will run when your computer is unlocked:

Start > Administrative Tools > Task Scheduler In Left top corner select Task Scheduler Library click Create Task in the Right top corner in the Create Task dialog: General tab -- provide a name for your task Triggers tab -- click New... and select On workstation unlock Action tab -- click New... and click Browse... to locate your script Conditions tab -- uncheck Start the task only if the computer is on AC power