So, a quick update on what I ended up doing - I took everyone's advice and sort of came up with a "hybrid" design using both lumber and foam. I built a perimeter frame of 2x6's around the sides and rear with a 2x3 at the front and then filled the middle section with 1.5" Foamular 250 rigid XPS foam. Pretty happy with the results since I think it's relatively light, supports the entire floor, and also doesn't make me worry about compressibility of the foam since the largest load-bearing areas are supported by the lumber. I'll take a picture at some point ...
