Question and solution here: https://superuser.com/questions/402070/run-task-scheduler-task-on-demand-from-limited-user-account-in-windows-7

Go to C:\Windows\System32\Tasks find the related task and assign "read and execute" rights to the user you want to be able to access it. Be sure to assign to "current object only." Then the task will be visible and runnable from the limited user, and it will work if you saved your credentials in it and checked "run whether user is logged on or not."