I don't see the advantage if you've got to lug around the same weight of shielding material . It's hardly surprising (to me, anyway ) that it takes a larger lump of foam to provide equivalent shielding to that of a smaller lump of solid.
Syntactic foam has been around for 60 years. It's used for buoyancy blocks on subsea equipment like ROVs, and the subsea tools that I deal with at work. I don't have any love for the stuff. It's fragile and expensive.
I saw basic research on metal foam, mainly Aluminum, being done up close and personal. Strange stuff.
I was wondering if it vould be so when I read yours above. What if externally covered by a thin plating of the same quality giving it some mechanical rigidity? Kind of those containers of various kind with a simple structure filled with some type of polyuretane..(?) Even lifeboats are /were? built like that.