I think the most elegant way would be to create a LV DLL which will be called from your program. The DLL will accept the VI path, open a reference to it (using Open VI reference from the Application Control palette) and wire that reference into a property node with the Front Panel.State property which will have a "Minimized" constant wired into it, and then close the reference.
The only problem is that I never worked with LV DLLs, so I'm not sure how you export a VI to be a DLL. Search this site, or maybe the example finder, or read the help, to learn how to do this.
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