I would go to the very end of engaging every single possible tool. Having been naive to think that LH cares about its passengers, no matter what they pay, I have discovered that all they care is not one's loyalty but the profit of their Group. Hence, I do not care about the airline, either and pursue every single cent that they legally owe me. The usual scenario looks like this:

Passenger: Letter 1P with legitimate complaint sent
Airline: Nonsense reply 1A sent 4-6 weeks after saying that they are sorry about the inconvenience, but can't comply as there as absolutely nothing that they could do. Worst case: they do not even address YOUR case
Passenger: Letter 2 pointing to the nonsense from airline's letter 1A sent.
Airline: Apologizing for nonsense from letter 1A and finally addressing the issue, but reiterating that in this case as well nothing can be done.
Passenger: Letter 3P sent to the airline and to the EU enforcement body. The airline is informed about you involving EU agencies.
Airline: letter 3A apologizing about letter 2A and giving up to letter 3P. Compensation transferred, case closed.

Last year, I created a template for letter 1P, 2P and 3P, which, funny as this may sound, work like charm and all I need to do is to just change flight info. Addressing airline's BS remains intact.