The single hole per corner image is what I was picturing.
I hesitate with the 3 hole per corner approach because I expect the space between holes to be somewhat fragile. And if that is the only time your string is above the material, then that is the only segment supporting any real pressure. If you make that gap large enough, it should work pretty well.
The tile-like backing approach may suffer from the hex pieces peeling away from the backing along the edges. Real small hexes should avoid that issue. But since the tile is normally not on a constantly flexing subsurface, what works for them may not work for clothing.
If that is delicate, it is because the wrong material is used. They are using a rigid material and relying on thin sections for flexibility. I would use a material that is flexible and make the joints much stronger. Nylon would probably be my choice, though I have experimented with using hard materials with rubber-type joints on my dual-material printer, to quite interesting effect.
I personally think that this is a design where a 3D printer may have an overall advantage over a laser - if you can get the design developed correctly, a 3D printer will give you a result with much less assembly time than a laser cutter.
I spent some time thinking over this on the weekend and had a bit of an idea.
Following the button approach I would basically build large hex buttons and stitch them so they are sturdy on a piece of fabric. The holes would have to be fairly far apart so they would be sturdy on the piece and I would have to build a spacer to get them in the right spots as I had imagined before.
NOW to get the flat hex on top I would engrave a recess out of the center of a slightly larger hex and then epoxy it on top of the ‘button’
