You could try gluing canvas to foam core. Just glue it around the edges. Canvas does not have to be stretched super tightly in order to paint upon it. In fact, that is how floor cloths are painted. the paint is applied to canvas that is not stretched. Then it is varnished after painting. Both canvas and foam core can be cut easily with an x-acto knife. Or just paint right on the foam core. It has a very smooth surface for drawing and painting on. It is very light and can be placed onto a mini-easel on a table or shelf. You can also glue a small flap doodad on the back so the foam core will stand as a framed photograph would, all by itself.
Try using regular white mat board which you can buy at the art canvas store. Use Elmers white glue and a small foam roller. Cut the canvas about 1/4" larger all around. Apply the glue to the front of the mat, lay down the canvas. Sandwich between two layers of Wax Paper and weight until dry. Wrap and glue the edges, Sandwich and weight again.