I actually did have similar problems in CPH several times as an FTL flying LO and LH. I was denied access to business check in. I elevated the problem to the shift manager twice. With similar experience like you -- complete refusal. "Lufthansa just won't pay for this".
I wrote complaints to MM, receiving replies similar to the one above. Did not help. At some point I started to enumerate how many times I was denied promised services (in my letter emails to MM) and requested compensation. Not that I needed the miles. I more wanted to make sure that the answers do not come without any action from MM.
I received compensation and was assured that they discuss it with SK (as the handling agent). This has helped. For several times I was not denied the service.
Then I become EBG so I do not know whether the problem persists. But I learnt that applying pressure on the floor in CPH does not work. I try just a bit, and if this does not work I give up. Then I think one should apply strong pressure on MM. SK is their subcontractor in this respect, and LH/LO have instruments of pressure to use to receive the service they order from them