Once I had my four holes created, it was time to set it on the bottom. To keep it in place securely, I opted for an unusual product: spray Elmer's glue. Say what you will, but this stuff totally rocks. I use it for tons of crafty things and it works great with fabrics, woods, plastics, and more.
 

I sprayed it generously around the entire bottom side, and carefully around the edges where the fabrics overlap, then lined up my holes and pressed it onto the bottom. After working with it for several minutes to get it in the right place and eliminate any air bubbles, I have an official bottom cover attached.