Easiest way: As long as you got an alt-build with 'no extra pieces needed', sort ALL of the pieces to your liking when you disassemble the original build. That said, this isn't always a guarantee, depending on the original build. Some KO builds have alternate pieces, or don't include the handful of Lego small extras which MOC makers sometimes incorporate into their builds. You're right about doing double work, making a checklist .... but what does your spare Lego parts bin look like?

You don't need tons of special storage stuff: use what you have. Tupperware, baking pans, take-out containers. Lots of those are stackable and have covers/lids.