Josh,
If my measurements are correct yes, the background will be 36 inches, which leave 11 1/2 inches for the false bottom, the door is 36 inches, may have to tilt it a little, more concerned about staying within the dimensions side to side. If this does not work out, I plan to use the same design idea, but use GS and just carve it all with the hot knife, that thing works great for shaping and cutting without making a big mess.