When I took classes in this, the recommendation was to use thin cotton for the overall quilting and puffy poly for the trapunto'd parts. At that time, we used water-dissolving thread to sew around the designs, then cut away the poly from the back -- kind of tedious, as you have to be very careful not to accidentally cut the top fabric!
Since then I have seen a different approach. Basically what you do is cut the trapunto design out of the poly, spray the design on one side with basting spray, and then position on the wrong side of the top fabric. Eliminates the problem of accidentally cutting the top and saves a lot of time. Probably the best way to cut out the trapunto design from poly is to delicately iron freezer paper (with the design traced on it) to the poly batting, cut the design out, then remove the freezer paper. You have to be *really* careful about iron temperature because poly batting melts if the iron is too hot.