I'm sure theres a way to convince her. Did she agree once and then have you keep asking?
Er... I vote for either the giant storage box idea, or maybe go through the ideas everyone uses about getting a hamster, except try them about getting the hamster a bigger cage?
You could always make it clear that you saved for it, or even buy a big cage in a pet shop locally and mention that she only stopped you getting the ZZ2, not a bigger cage lol
Am sure there are ways around it, they just need to be found...
Or yeah! Giant bin, easy and cheap. Making loads of small holes shouldn't be too hard either