I'm not sure I fully understand your use case, but i'll try.
What you can't do in story (at least for now) :
- apply different filters each time you open a story
- cycle a specific filter on a sheet inside a story
Taking that into consideration, Story won't help you achieve your goal.
Maybe design a sheet, and print the whole page, after you make selections.
It's not ideal, and very tedious work, but I don't see any other way to achieve what you want.
By the way, in Qlikview, you have the option for a Report, that can do the above, so I hope Qlik will work on that in Sense as well.