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If the singleton points share common attributes, you could convert them to mulitpoints, which would reduce the record count, and therefore the size of both the .shx and .dbf (you might get a little reduction in the .shp, or it might increase a bit, due to the cost of storing the envelope, depending on the vertex density in the multipoints).
.shxfile is fixed-width (with length(100 + 8 * nrecs)) and required. See gis.stackexchange.com/questions/348557/….shxand.dbf(you might get a little reduction in the.shp, or it might increase a bit, due to the cost of storing the envelope, depending on the vertex density in the multipoints).