She will always behave differently in a play bin, out of her cage, than in it - because she will see it as a play area and will know she still has her cage to go back to - her home. If you moved her to a bin she'd be wanting to escape.
I think it's too narrow for a cage. She will run on her wheel more at night in her cage - they settle into that and don't use it as much at other times. But in a play area they play.
I think if you wanted a bin cage for a dwarf hamster, you'd want something the size of this Rub bin, which isn't cheap. If you want to upgrade her to something bigger then either that maybe, or the Barney cage?
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Both the Barney and Alaska are about the same height and both have fall risks for a dwarf hamster, so if she is climbing bars then it would need setting up to avoid fall risks.
35cm is far too narrow.