Thank you, that is a very nice construction!
Starting from this position:
Black first achieves W+2:
(ending in gote - only playing part of the sequence seems like a bad idea for black)
White first achieves W+3:
(ending in gote, because black does not have to play 2 immediately)
So this does have all the desired properties - only drawback being that it relies on the global ko threat situation. But perhaps this is what Bill Spight had as well, in which case this might be as close as we can come to a “real” button.
If either of the two ideas from @hexahedron could be realized, I think those would be less sensitive to kos and ko threats. But they both seem quite difficult!
On the sensei’s page for one-sided dame there is only one example given:
What other variations are possible? I’m pretty sure I’ve seen at least one different one before, but I can’t remember it off the top of my head.