Both LH and LX are not state of the art any more in business class. You could still book a seat in the middle pair of seats to avoid the problem you are describing.
The 321 on LH and LX just have their (pretty bad) regional economy seats with an empty middle seat. Avoid these. LX usually has a widebody with lay flat seats, but they substitute.
You are not saying what kind of RTW ticket you are on (it cannot be one of the Star or Oneworld variants). Is your TA building this based on one ways?

If on separate tickets:
You could also look at TAP via LIS. Lay flat on both legs (at least when they have the 321LR on the TLV flight, free stopover in LIS, usually good fares. Service and lounges in LIS are just ok, not memorable.
SQ also has a flight FRA-JFK with very good seats and great service, but that leaves too early to connect same day.