@Tigerskin I think you are missing a step. Since plants only give a cutting from the base plant, you would need to then grow some of the ones you created by grafting (plant a harvested fruit) and take cutting from the 'pure' plants.
So you would take that dragonfruit, plant it, and graft a snapdragon to it. This will allow it to produce cowplant berries. That is the desired end result, to have dragonfruit, snapdragon, and cowplant berries growing on that plant along with the bonsai buds. You're right that you won't be able to take cutting from a cow plant. You should be able to take cutting from bonsai buds (not the bush you trim) after you graft strawberry and daisy together, then plant the resulting bud. As far as I remember, anyway. Been a while since I gardened.
I guess I should edit this within the introductory paragraph to point that out, but would like to hear from you that this fixes it for you.