I cannot be absolutely certain, because the entire conversation was in German and German is not my strength, but I believe they accepted the document from the Minnesota DMV and I should receive my Swiss Driver's License as well as my US Driver's License soon.
Either that or I am being deported. It's kind of difficult to say, what with me not speaking much German and all.
Thank you for everyone's advice and support. And, of course, my parents' emergency visit to the MN DMV office plus my brother-in-law scanning and e-mailing me the document.