Capturing images on fast moving conveyor belt
Try reducing exposure time if your camera allows that. With the short exposure time the conveyor belt should be well-lit.

If that is not enough, a common solution is to use a flash which is synchronized to the camera. If a sync signal is not readily available from the camera and you can't trigger the camera with your own sync signal, you probably could find a pin on your prototype camera's circuit board that carries a signal you can sync the strobe light to.

Without synchronization, setting the strobe rate to about 2/3 of the frame rate may be a reliable solution that gives good frames separated by at most one bad frame. Whether that works depends on whether there is a rolling or a progressive shutter and on the exposure time. Other ratios closer to 1 might work better in case of a rolling shutter and/or a short exposure time.

About DSLR cameras, you should use a camera that uses an electronic shutter in the mode (still/video) you would use. Cameras do not normally overheat but a mechanical shutter is only specified for that many cycles.

Also line scanners are a possibility, but I have no idea if those are commonly used. Think of a flatbed scanner where instead of moving the scanner head the target is moving. Those never blur the image but require bright illumination along the line being imaged. That may be easier to arrange than illumination over a large area, but it's a more exotic solution than a normal area camera.

More of an anecdote: A synchronously rotating prism with parallel opposite sides, located in front of the camera, can approximately compensate for linear motion.