Surely they must exist? I do remember the Sandhill cases from the 60s, which held each denomination for a particular year. However, if you picked up a few of these cheap for 60s dates (you will want a 1965 which will hold from Crown down to halfpenny, and a 1966 to get the standard halfcrown down to halfpenny), you could adapt them to hold denominations for each monarch from 1816.
Otherwise you might have to bite the bullet eventually and get a cabinet, or if that proves too expensive, you could go for Lindner trays instead.