And the answer is....

[rhu]

At shul this morning, I was able to ask the gentleman who led Pesukei d'Zimrah on the second day of Rosh Hashanah about the nusach for Shir ha-Yachid. The answer is that it is the custom at our shul to use the melody of Akdamut for Shir ha-Yachid before switching to the regular nusach la-Yamim Noraim for the rest of Pesukei d'Zimrah.