I won't say it is impossible, but there is one very significant issue that is going to arise while cutting, trimming, etc. And that is that you are gonna get a lot of pieces chipping off. It will probably be difficult to get a smooth surface.
I'm not sure what you would be using this for, but if it's for trooping, you're gonna have a problem if somebody bumps into it.
you can always fiberglass over the foam I know someone who's made a set of armor that way and it turns out really well. I was going to make my JP that way but I'm fiberglassing over cardboard instead now following WoF's templates.
Polyester resin will eat this type of foam, so use an Epoxy resin or protect the foam before fiberglassing with water-based polycrylic (i.e. Minwax) , several coats. People have been using this foam for building rc planes, then some fiberglass them. I have seen them use water-based polycrylic (i.e. Minwax) with fiberglass cloth also. I was also planning on building a jet pack using the pink foam and then fiberglassing it.
Here is a site I found. This guy is using pink foam and fiberglass to build things. http://www.fiberglassblades.com/Process_FF.html