A good and unusual question. Maybe best posed to people who keep birds in captivity (and presumably in colder places than the species normally occurs in). Also those who watch roosting waders for a long time may well have thoughts on this. I suspect from watching canaries and finches that they don’t change their legs for sleeping on (at least not often enough for what might seem necessary to control heat loss. But I have no data on that. One thing I do recall though that if one of these birds is sleeping on both feet it is probably sick or distressed and likely to be dead in the morning.
Philip