Not sure why you don't get everything in the export. Exporting is the prime way to make a join permanent (not a great practice IMHO). Are you using feature to shapefile or geodatabase table tools or the Geoprocessing tools? If you export from the attribute table menu instead, you only get attributes, not shapes.
As to the main question, you can only edit Table 1 (the first or "left" table in the join) in any join. So you have to join from the table you want to edit to the polygon feature, assuming you want to see its contents when you edit the table.
Also, in order to edit a table, it must have an OID, i.e., it needs to be converted to an ArcGIS table using tools such as Import or Excel to Table.