Most pinbacks are made with non-ferrous metals and wont stick to a magnet.
Don't use foamcore use insulating foam from the hardware store; similar to the foam in disposable coolers. It comes in multiple thicknesses, 1/2" to 4" and is super inexpensive, extremely durable. I still have the same boards for over 10 years now. Just changed the fabric for the third time.
What's nice is that they come in a 3x4' sheet and you can customize it any size you need. With a hacksaw blade or kitchen knife. If you want a real classy look fit the foam into old picture frames.
Speaking of which, you could use old frames and staple any number of fabrics to the inside. You could even use screen and different types of wire mesh depending on the look you want. If you did this option you'd slow down thefts too. Since you could pin them.
I'd do a small display and then leave the pinbacks in gift boxes in trays. You can make trays by repurposing used frames. I use multicolored boxes, purple and green happened to be on close-out and they make for a stunning combination when viewed from above; with the white batting softening the whole look.
People love a mystery, exploring, finding hidden treasure. Customers tend to linger searching for the "perfect" one. It tends to keep your booth full.