You have to realize that this is not an inertial reference frame. Basic special relativity is not sufficient, we need to consider the rotation.

If we choose a reference frame that is external to the centrifuge, the center light is not moving, and the edge light is going near light speed. The problem remains to be your timer. It is halfway between the lights and it is moving as well.

Now we need to pick refence frames that are along the disc, and this does not help your cause. We'd almost have to pick a timer that is a radio beacon some distance away from the disk and on the normal of its plane. If it was far enough, we could say the signals will be synchronized.