No. Because the video recording is being taken relative to the crew and being sent. The transmission itself is not relative to the ship. You would see a normal video.
If you were OBSERVING the ship itself it would be doppler shifted. But you’re watching a video from inside the vehicle travelling at light speed.
But you are telling me the frame rate information contained in the carrier would somehow not be Doppler shifted along with the carrier?
I do radio engineering for a living. What do you do?