Probably not. A monochrome camera generally does not save meaningful power compared with a color camera. Most image sensors are fundamentally grayscale sensors; color is usually created by adding a color filter array. So removing color does not change the basic sensor technology enough to give the kind of battery-life gain you’re looking for. Lower megapixels also usually won’t make a big difference in power use.

For a week-long, low-maintenance monitoring setup, a trail camera or some CCTV-style cameras are a better fit. These are designed for long unattended use, often include motion detection, and many can work in darkness using infrared illumination. They also often avoid wasting power on large built-in displays.

If there is no light, you will need either IR illumination or another light source unless you choose a camera with built-in IR capability. If your goal is simply detecting changes in a trap, motion-triggered capture will usually save much more power than trying to find a grayscale-only camera.