How to move folder of SQL Server LocalDB instances from the default location?
See my comment above. I used Process Monitor to trace the registry access and found it was getting an Access Denied while trying to write:

3:16:40.8405491 PM sqlservr.exe    8756    RegCreateKey    HKLM\SOFTWARE\Microsoft\Microsoft SQL Server\MSSQL11E.LOCALDB\MSSQLServer   ACCESS DENIED   Desired Access: Write

I got around this by opening the Registry Editor and giving Everyone full access to HKLM\SOFTWARE\Microsoft\Microsoft SQL Server\MSSQL11E.LOCALDB\MSSQLServer. Then I changed the default location and removed the Everyone access once it was done. This made the following extra keys that weren't there before:

DefaultData

DefaultLog

BackupDirectory