The broody hen has been a little more cooperative recently. Yesterday she was off the nest and waiting to exit the coop with the other hens. She went through her dust bathing and eating routine. She rejected the rooster's advances again.
Last night she got on the nest for a while, but later perched on the roost. Yay!
I wondered if she was pulling another sneaky trick, but today she was out with the hens again today.
Tentative verdict: cooler cage works, but it takes a week or so.