The only thing that has ever worked for me is to secure them with double sided carpet tape.

I don't think you can attach them. The force of walking on them will either rip the pins out or rip the carpeting.

The only thing you can do is resign yourself to straightening the carpet on a daily basis.

I thought of double sided carpet tape, but was afraid it might leave a sticky residue on the permanent carpeting.

I thought of even nailing them down with a large headed nail and a washer! :) Sub floor is just plywood.

You know those 3M hooks that remove without leaving any damage to the wall? They have them like that only not hooks, but with velcro tabs. They are marketed as picture frame attachers. But I think you could use them without worrying that they would leave a residue on the carpet. You'd put some of them on the carpet and some of them on the runners, and then they snap together.

You can separate the velcro tabs when you want to wash the runners. I don't think you'd have to remove them, but I'm not sure how the ones on the runners would do in the washing machine. You could always remove the velcro tabs by pulling the glue strip down and put fresh glue strips on the velcro strips on runners after washing. They are pretty cheap. They sell replacement glue strips separately so you could reuse the velcro parts.

You can get underlayments that are supposed to stop carpets from walking. Anything that actually attaches the two carpets will damage the wall to wall most likely: pulling at the the loops of the berber and fraying them or actually pulling the loops out, eventually. Occasional spot cleaning of rugs is probably less damaging than trying to attach runners.

Palimpsest, what do those underlayments look like? I have some rubber matting, in kind of an open waffle weave, I bought for a throw rug on a bare floor. Is carpet to carpet different?

A thick rug pad may hold it in place. You want one thick enough so that it gives very little, if at all, when you step on it.

Try the thick felt type or the ones that can be used on either hardwood or carpet (one side is rubber, the other is a rough texture).

In place of the rugs you could use the clear vinyl mats that have the spikes on the bottom. It's the same as the mats made for office chairs.
Another option is to use a heavier, rubber-backed rug.......like the ones you see in entryways.