I had to laugh at hubby's stash in your purse! I do that to myself. I have a little flashlight (4x1, heavy!) in case I'll need it, etc. I used to carry a metal tape measure, but now have a paper 3' one I got at Jerry's. The nylon stuff sounds good. Many people use "pet screen" which is a strong plastic with holes in it. There is a place here that sells it in bulk, and also gives away scraps but I haven't been there yet.
We haven't put the feet on yet, and I haven't finished the purse yet, but I will post pics when it's done.
I like your idea of foam core, but don't know if it would be sturdy enough with the feet I'm using. I want it pretty stiff to keep the fabric off the cart. Like I said, this is a learning experience.