It took me less than 30 seconds to find a local Athens phone number for LH. +30 210 6175200. Call it. Do not email.
EC 261/2004 does not say anything about "next best" All it requires is rerouting under "comparable transport conditions." There is no precedential case holding that rerouting on another carrier is required. Nonetheless, you have three options: A: cancel for a full refund; B: reroute under comparable transport conditions, or C: reroute at some later time.
If your flight is within 14 days of the notice, you may also be entitled to delay cancellation depending on the ultimate changes which must be made and the delay at your final ticketed destination.
As a starting point, you should do the research. Write down the options and then call LH. Ask for what you want by providing the information. It is a lot easier to get what you want when you provide the details and make it easy. LH will need to have an interline ticketing agreement with the carrier you propose.