I make my own targets, when I see a box about the size of a target I would like I stuff it with a mix of cardboard and plastic "wal-mart" bags. Usually I'll layer in at least 20-30 layers of cardboard to the front of the target, personally it is cleaner to pull the arrow out of the target this way, and back-fill with the plastic bags. I then tape on the front a sheet of marked cardboard with a crayola bull's eye and every few hundred shots or so I replace the front sheet.
Sorry that answered your subject line but isn't a solution to a 4' round portable target. I have considered making custom carboard "boxes" that are much larger, in the scale you are describing, but the weight would be prohibitive with how deep I make them (10-12 inches?). However, you could probably make one much thinner. I would think cardboard and plastic bags would have more stopping power than styrofoam?
Does the cold negatively affect the foam you sprayed in between the styrofoam, or just the styrofoam? Any chance you could skip filling in with styrofoam and just use several cans of foam?
Maybe scratch that idea, based on your website example that would take a couple dozen cans or more, right?