Perhaps if the shot cups were carried in a LOADING BLOCK- thicker, of course, than what you would use to hold round balls-- they would be easier to handle, and load into the barrel. I would only want 5 or fewer such cups with shot in them for any given block given how much weight the loaded block would have, but it would be a great way to help quickly reload a fowler, or shotgun in the blind.
Any way you can work it so you don't have to carry loose shot in a bag, or flask, etc. to load your gun has to be okay. I was thinking that if you made a loading block, that has a bottom to it, sliding back and forth in a rabbit groove, so that the heavy shotcup does not fall through the block, you could find a way to seal the mouth of the cups, other than using tape, to hold the shot in the cups while being carried.
I just like your idea of loading the shot in the cups before you go into the field. My field experience, to date, has been limited to loading 12 gauge cups, as I have not yet tried it in my 20 gauge. But, I have large fingers, too, and I suspect from what I have already experienced working with my 20ga. that I will have similar problems like yours when it comes to loading the smaller gauge. Thanks for the candid appraisal of your efforts to date. I don't think you should be completely discouraged, however. You just need to solve some problems you didn't see coming at you to make this work.
Ideally, a light, clear, plastic tube with the inside dimension equal to the diameter of the OS cards would make a good way to carry a loaded shot cup, with the OS card(s)? keeping the shot in the mouth of the plastic tube, and some kind of bottom holding the cups from falling out. You would load the shotcup into the barrel the same way you would use an In-line percussion capper.
Maybe, if I can find such plastic tubes, I can glue them into a light,wood block, drilled to the OD of the tubes, putting 3-5 holes in the block, with the sliding bottom on it, so that the bottom is slid back to allow the loaded cup to be pushed through into the muzzle of the gun( with a short starter?)