I would remove the strap, use small plastic clips to clip onto wall of cage.
then you can zip up and take them out much easier.
Sometimes you just have to put on fleece hand protection, grab that little guy and put him in bag, but it can be hard to do, I find a larger fleece piece is better, almost like cover them up.
but the ability to grab them can be important, if you open pouch to give them a treat and one jumps out, you need to be able to grab them up again.