This sounds like a challenging customer issue that may require consultant time or some time with our Analytics support team to assist. You could open a ticket at https://support.tableau.com but they are very limited on time so one other option is Tableau (or another firm) consulting services? This would really take some digging into with the cube data.
Only options that I can think of without a look at the workbook - check out Sheet Selection on Steroids or a tool to extract the data needed from the Cube into a more friendly db.
Oh and the type of cube would be helpful to share. There are differences that our audience will be more knowledgeable about if you share the specific cube type.