Passengers with special needs have various rights when they start their journey from the US. There is a special page on LH.com which spells out the details:
https://www.lufthansa.com/us/en/trav...h-disabilities
Unfortunately there is no way around paying for a seat assignment as all passengers in economy have to pay for advance seat assignments on longhaul flights. You could try your luck by calling the medical desk and explaining them your predicament.
Lufthansa medical desk
If after reading the information provided here you still have a specific medical question that has not been answered, please feel free to contact our medical desk in New York. The Lufthansa medical desk is open Monday through Friday, from 9:00 a.m. to 5:00 p.m. EST at the following number:
1-516-279-1768.