Unfortunately, the Nikon 600mm f/4E FL ED VR does not balance with the D810 and the foot replacement LCF-13 by Really Right Stuff nor with the Nikon stock foot. The LCF-13 worked perfectly with the previous model 600mm f1/4G ED VR.
After some research I ordered the foot LF-N604FL by jobu design. It is 4 cm longer than the Nikon stock foot and 2cm longer than the new LCF-18 by RRS, which is too short according to this comment. Now, my setup balances fine horizontally, but vertically the lens sits (still) too high with the gimbal by Dietmar Nil.
To balance my setup also vertically, I needed a different gimbal head. The WH-200 Wimberley Head Version II would let me compensate a height up to 12 cm. I went with the Jobu Heavy Duty MK IV, which is a bit lighter, but only gives you about 10.5 cm (my combination requires at least 10 cm).
You may wonder why the foot has such a high profile. I guess this is owed to the longer lens hood, which reaches further back when stowed away.
But I don’t really see why that was really necessary. I think it wouldn’t matter to have the hood cover the foot as shown in the mockup below: