I eventually came across a used satellite dish, and with a little bit of modifying this turned out to the be best option for a Captain America shield.
If you are interested in making one of these, you may have to do a bit of searching and asking to find a used satellite dish. I've seen them show up at thrift stores from time to time. Keep your eyes open, and you should be able to find one.
I have a separate instructable on how to make a flyable version out of cardboard and duct tape, which you can check out here: Flying Captain America Shield.
Thanks for looking!
Step 1: Disassemble the dish
Step 2: Draw circle
I was worried that a circle cut from the oval-shaped dish would look warped. But once it was cut out, the circle looked perfectly fine.
I eyeballed where the center of the dish was and put down three or four pieces of masking tape, which provided a place for the needle of my homemade compass to bite. I then drew as large a circle as would fit within the back lip of the dish.