Filing a complaint with either the DOT or Transport Canada will not get compensation as this is not a denied boarding in the legal sense but denied transportation. If this had happened ex-EU IDB might have been due. Personally I'll never understand why IDB laws in North America don't include compensation for denied boarding in situations where the airline was in the wrong.

As no compensation is due and the passenger was not out of pocket for other expenses ( which the air might be on the hook for) I would right a letter to both LH and AC explains the facts. This might get a small customer service goodwill gesture in th form of miles or a voucher.