The Belyi-(1) families with Count > count = 5 are precisely the ones that appear inside Belyi-(2) families.
The Belyi maps with Count = 5 and count < 5 are precisely the 5 plus 1 maps that appear in the Count=5 table but not in the count=5 table.
So the easiest way to construct the first column of the count = 5 table is to construct all Belyi maps with Count=5 and then delete the 5 plus 1 cases with count<5.