No, the beam of light would not "bend" or even appear to because you moved the source. It can do so in a gravitational field but that is another topic. The suggestion that if you tug that it sidesteps the rigidness is also incorrect. The modulus of elasticity dose not change for a given...
Even if we assume there is an ideal stick that is perfectly rigid if any part of the stick approached the speed of light it would gain mass as all things non light do. Thus making it impossible to go faster without more energy which in turn adds more mass. Group velocities can travel faster than...