There are a couple of things you can try. First, if you hover over the labels you should get an icon indicating that the labels can be moved. If you move the labels to more appropriate places to avoid confusion, you may be able to arrange them so that the map cleanly displays all info for your export.
Additionally, if there is a label that can be easily shown as another viz type, you could always combine the map with a small bar chart (or something else) on a dashboard and export that so that you get both vizzes with all your info displayed.
Hope this helps,