For that purpose I use the freebie backpack objects that come for free with shoes.
My small Samsonite suitcases each came with one, with the intent that you'll pack a pair of shoes in it
My last pair of Rapha shoes came with one
My last pair of Giro shoes came with one
I see them at the Japanese import store for ~ two dollars
It's a nylon drawstring bag, but there are two drawstrings and each drawstring goes to a lower corner of the bag, so you can wear it as a backpack. If I Google "nylon string backpack" I see vendors selling them for about a buck apiece. Amazon has a package of 16 for $22.
The one I bought from the Japanese import store definitely packs to smaller than 2" x 2" x 1".
Bill Lindsay
El Cerrito, CA