Try a light iron-on interfacing on a square, and was as usual. You should be able to tell from that whether you can live with the stiffness that remains - some are fairly soft.
The bigger problem remains - how are you going to quilt it ? Anything satiny is going to be a nightmare for machining, especially on the back. Hand quilting would almost require staying with the grain as much as possible.
You may be able to piece a center medallion for the school logo, and the rest of the back in a more practical fabric. Others have already suggested polished cotton, one of my favorites. It looks good and holds up to wear. Down the road, it will still be looking good, where poly could be in shreds.