All you need, is a valid apple Serial from a real mac. In some cases, they don't even want to know the serial. And if you don't have any, just tell them your actually not home and you just have copied the Support Code. Anyway, the Serial is only needed to associate your case with a device. Explain, that you get an error while trying to log in to iMessage. They will ask your for the Code (Something like 9378-3847-3904). After that all should work!
If possitioned correctly, you do not need a SN at all, read my post here about this.