@johnirvine That's good information. It's unfortunate *G are not treated equal; I guess TK E+ is closer to SEN/HON qualification which might explain LH behaviour.
From what I've observed, A3 status is only recognized properly during the on-line checking if the card is already added in the booking system 48h before the departure. Otherwise, you only get an A3P on the ticket (aka no status) and you can only override it at the counter; neither the technical support nor the reservation office could help me (we see the card in the system but we don't know why it's not reflected on the ticket).
Fwiw, 2 airport agents and one representative over at the reservation center confirmed that the issue occurs a lot and most of the times with A3 cards.
All things considered, it's disappointing to see LH doing this kind of shenanigans.