There is definitely a problem with the way the application is installed per machine. What you're describing indicates that the application is missing some information when running under the user account.
A couple of things to try.
(1) after the application is installed in Admin mode--launch the QualityMonitor and run the application using the user log in credentials. This should tell you exactly what happened or get you started in the right direction.
(2) If QualityMonitor is not available, use Filemon and Regmon to monitor the application as it launches.
I can't tell too much based on what you've described, but my hunch is that there is a HKEY_CURRENT_USER key written when the install runs in admin mode that isn't available when run with the user account. You may have some luck searching the .msi for any HKEY_CURRENT_USER registry keys and manually adding those in the user account to see if the problem goes away.
Hope that helps,