Some bags that I've had, had cardboard in the bottom. Some had iron-on heavy weight interfacing, but those shrunk, leaving a buckled appearance. (I suspect the fabric wasn't prewashed.)
Other things to try include template plastic, or plastic canvas, either of which could be covered in fabric.
All of these choices except the cardboard and your cardstock paper are washable. Cardstock wouldn't add much structure.
I'm thinking of bottom stabilization. If you want overall stabilization, I'd use the weight of interfacing that seems right to you, probably medium to heavy. It could be sew in or iron on.