The wires ended up not working very well - however, they did inspire a slightly different solution. Instead of wires, I now fill up the slots with a solid layer - but only a *single* layer, instead of the full height, which is just enough to provide extra rigidity.
I've also slightly increased the thickness of the whole thing again, because it was bending at the hooks too much, risking breakage under weight. I've also modified the end hooks a bit; they are now actually hook-shaped, and thinner in some non-critical places.
The end result is a plastic cost of 1.65 meters; only slightly higher than the empty-slot version, which was about 1.55 meters. The original design was 3 meters!
The design seems about perfect now; the hooks only engage with the rim of the bin once it's under weight and the hooks are pulled towards it, whereas without weight there's enough margin that you can take off the whole net without getting stuck on that rim.
And it's working quite well with some stuff stacked on top of it! And a lot more stable than my previous corner design :)
So far I have 5 bins stacked on top of each other on my desk, with the topmost bin containing a few heavy boxes of screwdriver and drill bits. The tower still seems perfectly stable! And I'm not seeing any kind of wear or fatigue marks on any of the stacking nets.
(These are printed in crappy PLA, to be clear, so I'm testing pretty much the worst case in terms of mechanical strength, relying entirely on the design)