Use non-rechargeable alkaline batteries. The Eneloop AA is nominally rated at by the manufacturer at only 1.2v; the Energizer AA is rated at 1.5v. That's puts the Eneloop at 20% below the desired nominal voltage.
The motor grip powers everything on the camera; you can remove the on-body button batteries.
I find that AAs in the motor grip last for quite a long time before they have to be changed. The manual specifies alkaline-manganese or NiCd batteries with a total output of 9v (6 x 1.5).