As far as I know, WeatherlinkIP does not directly upload to Wunderground. If you tick the box on the Weatherlink IP Configuration page, it will upload to weatherlink.com. Weatherlink.com in turn handles the upload to WU, if you have the uploads configured there.
Did you mean weatherlink.com instead of Wunderground? If so, does the battery-out/power disconnect recover it again until the next 3:15AM interval?
Are you running DHCP or fixed IP on the WLIP? I would recommend checking the lease time in your router to see if it has a short one-day lease time.
I have a station that uses WLIP->weatherlink->WU and it uploads just fine.
Help us understand the problem a bit better and I am sure that others will offer suggestions as well.
Greg H.