The only way that you can export to an Excel file that truly contains hidden columns is to build a summary that includes those fields, but on the appropriate tab for the field's properties, set the hidden checkbox. Then export that summary to the Excel file.
For Item and Measure fields, you'll find a Hidden checkbox on the General tab.
For Key fields, use the "Hide the key column" checkbox on the Display tab.
This may take some trial and error time to get the desired layout just right, but it can be done successfully.