Two things about LH these days:
1. Fuel surcharge is going to add $500+ to the award ticket;
2. Phantom F seats often show on AE and other search sites, but LH is releasing very few F award seats in advance any more, even at the last minute.

If OP knows the dates wanted, then make the booking when the window opens (is AE 330 days out?). Also, while not up to the standards of European and Asian carriers, consider UA F to Europe. The hard product is pretty good, but most importantly there will be no fuel surcharge (yet).

Business class will be more prevalent and increase your choice of carriers. I hate the UA business class seats/cabins so don't recommend UA (unless you can get a two-cabin CO plane) and suggest US. If you can't get a direct flight to FCO, you will have to connect through a European hub which will require a flight in European business (even if you are booked in F across the Atlantic). Also note that AC often flies one of its non-ExecFirst planes into FCO so it will not be true business class.