It doesn’t seem like text labels can be hidden with the Invisibility and similar traits that hide or switch the piece’s picture. Effectively, it doesn’t make much sense to assing such traits to a piece with text label (and vice versa). I first noticed the problem in the Babylon 5 CCG module, where some cards have text labels that appear regardless if they’re face up or down - so it’s sometimes possible to guess the type of the next card in the deck without drawing.
Would it be possible to give text labels an option to follow the visibility of a chosen layer?
This is a bug in the module. The order of the traits in each card/counter is vitally important. The problem you describe is because the text label trait has been defined below (outside) the Invisible trait. Trait’s only affect other traits that are defined above (inside) them on the trait list.