The Lee Load-All works fine for slug experiments although I will agree it ain't much for other uses. The easiest, cheapest, and generally the best approach is the cast round ball. Lee sells a .575 RB mould cheap and that will fit right into a 20 gauge shotcup. If you get lucky, you may be able to find cast or swaged RBs of that size at your local gun store for the muzzle loader crew. (I checked. Hornady sells swaged .570 RBs. That's close enough for this.)
That ball weighs right about 5/8 oz, but you will need to put a little filler and a bit of paper on top to get a crimp. Grits will do just fine for the filler and bring the weight up a bit so use 3/4 oz data. If you have to use 7/8 oz data, you might add just enough fine bird shot to get your total weight up to that. Three #4 Buck would do it, too and should let you skip the grits.
My work has been done in 12 gauge, but the principle is the same. The shotcup acts as a patch to make the ball fit the choke and drops off once out the muzzle.