I do not have Weatherlink Live, but if it uses a DID number to identify it on the network, it's probably similar to my situation a few years ago. I upgraded from an old USB logger to Weatherlink IP (before the debut of Weatherlink Live). After talking to Davis tech support, I was left with the persuasion that data cannot be merged. As I recall, the Weatherlink system stores data for each datalogger DID. A new datalogger DID = a new station file in Davis' data storage.
I set up a new station profile with the name I wanted to use, and renamed the old station with a name that indicates it is a archive of weather history (it is always offline since I'm not using the old datalogger). I'm using my Weatherlink account to relay data to Wunderground from the new station. Don't have Davis delete the old station if you want online access to the records.