You might have a problem with the zip tie coming off at the bottom because the box is gradually smaller. You could use a couple zip ties across the top of the box and connected to the zip ties wrapped around the box to hold it there better. I say this because you mentioned not being able to tighten it around the lip and I remember now I had the same problem so I used zip ties across the the top of the box. When I used my cages then I just used small zip ties attached to the cage and then attached to the zip ties around the box. Hope I haven't now thoroughly confused you.
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In the spring at the end of the day you should smell like dirt
~Margaret Atwood~
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