@Holbs you are never going to hide the camera's IR at night, but you may hide it during the day.
You may have already thought of this but when yhen you get around to building your bird box I have a suggestion. The FOV of the 6mm is about 57 degrees so you can mount if back from the hole at the front so it is in shadow and less likely to be seen. Obviously that raises the issue of IR reflection so you don't want any vertical surfaces forward of the lens to reflect anything back. My suggestion is to build a short tunnel.
You could use flashing or something flexible to make a tunnel the exact size of the outside of the face of the turret. Some quick maths tells me that you could have it at least 50mm long (2") from the front of the lens to the face of the birdhouse without cutting out any of your field of view. You could probably go deeper into the bird house than that but I just did rough calculations that absolutely wouldnt cut into your field of view. If you paint the camera, the inside of the tunnel, and the inside of the birdhouse matte black, and paint the outside of the birdhouse gloss white, to most casual observers it will just be a dark hole during the day. It should prevent the sun causing a reflection off the lens as well like the sunshade on a rifle scope. By night it will limit visibility of the IR to those that are already in the field of view.