An option:

In your pivot table try un-checking the totals you are using to create your Grand Totals columns, and in its place,

Click the edit icon for  Qtrs, and then "Create a calculated item".  in the dialog box, put 'Qtr1'+'Qtr2'+'Qtr3+'Qtr4'.  You can name it Grand total.

This may produce better results.

Did you change the position of your Measure labels?

Also, My last advice may not work.  The calculated Item will just total the percentages, not recalculate them.

In this simple example, the Column totals are correct:

But in your case, the complexity of dealing with unions is showing its ugly head.

Another approach gets into the real strength of unions.

You may have noticed the "Add a Result Column" option:

If you click on one of the "legs", then click on "Result Column", you'll see it.

This allows you to do calculations at the header level.

All formulas use column references saw_0, saw_1,saw_2, etc, starting on the left.

You can construct an alternate Percentage column there and see how that works.

Not knowing your query, lets say your column order is BIN, Week, Qtr, Name, Attempts, Completions

Percent would be 100*SUM(saw_5 by saw_0,saw_1,saw_2,saw_3)/SUM(saw_4 by saw_0,saw_1,saw_2,saw_3)

The BIN and name changes that you've created on the "Totals" queries will still work with this method.

Replace your old % column with this new one.

I know this is getting into the weeds, but your requirements are extremely complex/convoluted.

I'd have talked your customer into a simpler presentation, if I could.