Salty,
The large GP Balsa building board would work nice on a card table and stows well behind the couch.
Stay away from foam as it really is not stable enough holding pins (in my opinion) for a really good build. and a SIG SE fits well on the large GP Building Board. I am building a SE now and it has all been done on the GP large balsa building board. I use a cutting board to make my cut on (small piece of hard pine) and the balsa board cover in plastic to do all of the assembly. Plywood is awfully hard to get the pins into and I have tried the route before myself, another alternative to the balsa building board are 2'X4' acoustic celling tiles made from the same material as Homosote or Homeosote itself on a solid flat surface such as a hollow core door. I have yet to see plywood with out warps or twist with the exception of multi core furniture/cabinet grade plywood (cost upwards of $100 a sheet), and it still has some issues with twist and warping.

Good luck and have some fun with your building!

Regards
Paul