This is a very common problem with night vision cameras. Everyone wants to see as much as possible at night so they use long exposure times that brighten up the whole image. Then when a person enters the scene, their face reflects infrared light too strongly compared to the surroundings, causing the face to be overexposed and too bright in the video. And often due to the long exposure the face will also be blurred as it moves. The simplest solution is to reduce the exposure time until the face is clear. Moonlight mode is probably controlling the exposure this way.
If clearer facial recognition is the primary goal, you just have to live with the fact that the rest of the scene is darker than you would prefer. You can help the situation somewhat with extra lighting though. Ideally with wide angle lights that cover the entire field of view, with an emphasis on spreading out the light and not having it all focused on the center of the camera view.