Not an easy choice. I did both airlines in F on that sector.
Flight time:
I like night flights. Arriving in the afternoon vs. in the morning makes jet lag less painful for me. And I can sleep better on night flights. + for LH
Lounge:
The FCT is a clear plus for LH. As mentioned the CX lounge is pretty bad. You can eat in the FCT and then sleep for most of the fliight.
Plane:
The 380 is more quiet and has a higher cabin pressure. Also much less traffic in the isle and nicer toilets. Clear + for LH
Seat:
The CX seat is much more comfortable than the LH seat. More space (3 seats across in the 777 which is wider than the 380 upper deck), also better sleeping surface. The LH seat is very good, but CX is significantly better
Food/Service:
It is more a question of personal preference. CX F food has deteriorated somewhat over the last few years. LH sometimes has pretty good appetizers and even main courses. But it is airplane food after all.
Cost:
Assuming you use AA miles vs. M+M: Less miles and much lower 'taxes' for CX (at least before April).
Compared to QR F both LH F and CX F are significantly better. No experience with EY F.