Black-on-Black foam core is my material of choice for structure building. It is 3/16" thick. and it's inexpensive at about $3.00-$4.00US for a 20" x 30" sheet. Its available in most larger craft stores. Its also available in some office supply stores packaged as display boards, and priced higher there.
Unlike other foam core, Black-on-Black uses two layers of paper to sandwich a less dense, black foam. The paper takes glue and paint well and the foam does not noticeably dull the blade. Unless you're using a very dull blade, accurate, sharp cuts are easy and there will be no bunching up of the foam. This leaves a very clean, square edge for fitting and gluing.  One shortcoming is that it will warp if you let it. It takes paint well but only after the parts of your structure are glued together, and then there should be little painting (remember, we're covering this with a
veneer, not paint.)  Painting a large expanse of foam core is not recommended.