|This was suggested by someone who uses it for lining and supporting crocheted purses.
Use a premade fabric bag from the craft store, the canvas or denim kind meant for decorating with paint, etc. Turn it inside out so the good side becomes the lining for the bag you construct around it.
I haven't tried this yet, but I am thinking it will solve the dilemma of how to stiffen without overstiffening, and you can use the existing handles, or cover them, or cut them off. These bags come in a variety of sizes and are priced very low.