No foam? I'd say foam would be a pretty good choice. You can get those little knife thingies people use for carving details into things and make the lines of the puzzle...hopefully they'd already be selling a pyramid shaped piece of foam...and as for the ring, buy a smal foam ring or plastic one and hot glue....and as for the eye of it...I'd buy a little bit of foam again, half it, and then try to carve the eye out with the little knife thingy...i forget what they're called...not quite knives. Do you know what I mean? And why isn't the foam idea working? Too brittle perhaps?
This might be a ghetto suggestion, but you could always buy some plastic canvas (you use the plastic gimp rope on?) and just cut it to make a pyramid...hot glue some detailed cardboard or other material over it to give it the look...and then...voila - more or less. I'd go with something light.
I just thought it'd be really neat if the outside of the puzzle had tiny mosiac mirrors all over it...and that got me thinking that most likely, you can just buy jewels. Make the thing out of cardboard...and then use a hot glue gun to glue fake jewels to the outside of it (pretty sure they make long flat ones) - the jewels are always flat on one side. Some bling bling for ya.