I know a water colorist who wraps his art prints around the sides of 1/2" Gator. Front is glued as well as 6" long, blunted v-shaped wings around the back. Have never seen him do it, but he says he uses a thick rubber burnisher to flatten out and sort of stretch the wings. Looks great.
If anybody knows how to wrap thick canvas around 1/8" Masonite without developing a big ridge along the folded edges, lemme know.
About a year ago I built a sample frame as you describe, Masonite on the front with 1x2 Aspen board glued to the back of the Masonite as sides, glued with Miracle Muck. The bond has completely separated on two of the Aspen sides, even though I very thoroughly clamped them up while the glue was drying. The glue sticks to the Aspen, not the Masonite. I'm thinking there was some sort of process related coating already on the Masonite, maybe I had a sample of "tempered" Masonite which I believe does not glue well.