Please find an updated workbook attached. I tried to add the double axis map to your existent one but it didn't work. So I've created a new map, one axis with the filled map and the other one with a pie that represents the % of cats and dogs in each state. As it is only 2 slices a pie chart can be used without many concerns here.
I hope this helps you to complete your task.
US Dogs vs Cats V2 - rcalloni.twbx 532.2 KB