Speaking off the top of my head, have not done measurements, and this is strictly an idea in regards to Hero Character Cards.
I think it will have to be a new layout of an insert for your storage big box of choice. Let me see if I can ASCII Art this thing up.
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edit forum freaked out at my attempt at ascii.
Insert screen cap here:
Essentially, 4 horizontal spaces, for standard 40 card hero decks, and right next to it a vertical space big enough to hold the Character Cards/Variants for just those 4 heroes. Repeat pattern for as deep as your box/drawer is, and then repeat again until you hit all available heroes. Could be only 3 heroes, could be 5, I haven’t done any measurements for this. I also have no clue of what storage box/system would be suitable for this wider layout.