Looking at Lime Rock's published schedules (basically Pro and The Club), May 16-17 is open. NESCCA only has a National going on at Summit Saturday the 17th, and LRP probably only has car clubs those days; maybe dates could be swapped? This wouldn't have the potential for a double as July would, but a NARRC race would still be possible.
No idea about the school; maybe July 4 for the drivers that don't go to Pocono (if LRP would let us keep that after a move to May-what else are they gonna do with the unmuffled holiday?... unless they move their July 17 Test/Tune date there)...