The reason I asked if your drum was a 5x13 is because the barret Deems model was listed as that. It was part of a set that had a 12x20 bass drum, 8x12 tom and the 5x13 snare drum. The 5 or 5.5x14 versions were listed as the Buddy Rich model until he left to go with Rogers. Not sure that I ever heard of a 14" drum as a barrett deems model. Either way you have some great looking stuff! I think the piccolo is most likely the same shell material as the supra.
After looking at your pics again I think the Buddy Rich model had the super classic strainer. Not sure what your drum would have been called. seems like ages since I looked at an old catalog. Use it or lose it right?