I compost them. They die like everything else, and couch doesn't last very long in a bin. I think the antipathy to composting them comes from the tradition of open-topped heaps where they'd be able to establish themselves.
Any composting is good composting, but I don't like the idea of perennial weeds making merry in my wonderful compost - and it
feels like they are even if they aren't!
Are you relying on purpose-factory-made bins? (which I don't like enough to pay for), or does it still work with the classic 4-pallet cube with carpet on top?
If there's any uncertainty about whether I'm going to find 'strings' of roots in my compost, then I think the tiny effort of drying them to death is far less than the effort of finding and separating them out again...
Cheers.