And that gets a little more technical than I was asking, and it then crosses over into an entirely different situation ----
Valory Music LLC is not the label putting out music, Valory Music is doing that.
Valory Music LLC could "rent" their holdings(aka, copyright) to K-Tel, Rhino, or Time Life if they chose. That does not make them a label, Rhino would be the label.
So, by your example, we should be removing Valory Music LLC as the label simply because they are not labels. But we should not merge them with Valory Music because they are two separate entities.
*Please note, I used Valory Music, because Big Machine Label Group does own Big Machine Records, Big Machine Publishing, and BMLG Records --- I just wanted to keep any confusion to a minimum.