Just checked on your WU site and note that your current temp is -5.5 Celsius. I have had problems with my WS-1001 just recently, wind and rain gauge. Just could not work it out as the batteries appeared to be OK when I took my station down to check.
Suddenly realised that it coincided with a cold snap here in the UK. Checking battery outputs with a meter indoors versus cold operating conditions are not the same. Replaced the batteries and everything instantly resolved even in the cold nighttime conditions.
I know that there are slightly different arrangements for the powering the weather station between WS-1001 and WS-1002 but I hope this helps.
Ian