The webpage and driver are not linked in any way. The webpage is for jobs submitted on the webpage.
The driver will always print to the regular tray and the bypass is used when the job is programmed for what is in it, which typically is something abnormal to plain white Letter. So if you were to set the driver to print on "labels" and the printer bypass tray is loaded with Labels (Programmed as such) then the Auto select would work in the driver as long as Labels was set.