are a top tier customer in their FFP, in which case LH has a dedicated customer care centre. AF doesn't
Actually AF does have a specific handling process for Elite clients. Such claims are replied to by a dedicated team (well, at leat in Europe) in a matter of days as opposed to weeks.
There is a "brutal" way to do this: ask the credit card company not to pay for the AF ticket. Like that AF will "wake up" and deal with you, and hopefully give you the confirmation of the re-booking. You then can provide that to LH
Rather useless advise IMO. The Customer Care team will not know that you are disputing the charge with your bank/credit card company and this will therefore have no effect whatsoever on the handling of your claim. Furthermore, once AF has provided evidence to the bank/credit card issuer that the transaction was lawful (which it was), the charge-back will reverted immediately.
Looks like LH is to blame rather than AF. If you were initially given your 2 boarding passes, everything must have been OK in LH's system and with the new ticket (called FIM- Flight Interruption Manifest-BTW) AF gave you.