Pictures:
Box before any changes:

Red FunkyFoam:

To cover the bottom, I placed the bottom of the box on the Foam and traced a circle around it. I used a mechanical pencil, but I had the lead retracted. I scored the foam where I needed to cut it. No lead was needed. I had to score around it a few times to get a definitive line that I could cut. I then cut out the circle with scissors (NOT my fabric scissors!) and glued it onto the bottom of the box. I used a glue specific for FunkyFoam. It is very tacky, even when it is dry.

I'm having a lot of fun playing around with this. Next, I'll be creating a yellow circle for covering the top of the removable lid while the bottom of the quiver is drying. Then I'll cover the cylindrical bodies of the box and lid with red and yellow foam, respectively.
You know, it is really hard for me to tell the true color scheme of her quiver, because it depends on who the colorist is of any given issue. I've elected to go with a red body and yellow trim.