Sorry, not much help I know but FWIW:
AFAIK there is no schematic publicly available for the 6312 console (or indeed for most Davis parts). And, given that there is no logger or backlight involved, it's a rarely reported issue so minimal community experience to draw on. As per the table in the Davis catalogue, the average draw for a wireless 6312 with no logger or backlight should be around 1mA. So, presumably some component has gone relatively low resistance. But what and where is anyone's guess.