Thanks a lot!! I was able to get an AIX file out, with my WMS service as a background and some features from a FeatureLayer on top - exactly what I need 🙂
Its a bit weird for the users that you have to add a WMS to the WebMap in the old map viewer, open it in the new mapviewer ( where you wont be able to see the WMS background as its not supported yet) and do an export there. But I guess the users will be able to get used to that, and I guess WMS is supported soon in the new MapViewer.