Did you try this method? Using the above, I was able to customize the filename for all exports. However, I didn’t manage to set a filename that depends on the item I’m exporting. But maybe it solves your specific problem.

The issue seems to be the .toolbarBackground modifier, not the Map itself. I see the same problem when using a MKMapView, as long as I add that modifier. However, as you correctly described, Map will underlap the navigation bar by default and thus requires the .toolbarBackground modifier to achieve the desired look. Sadly I couldn't yet find a solution for customizing the back button appearance (which, to my knowledge, isn't possible without resorting to UIKit and appearance proxies) while using .toolbarBackground. Any pointers are much appreciated!