The captain accelerates and measures the doppler shift of the light from the star. Done. The star is not accelerated. The universe is not accelerated. The acceleration of the ship supplies the relative velocity difference. If the star had giant engines attached, then IT could accelerate and provide the delta v. Done. Once in uniform (non-accelerating) motion, all physics is the same in either reference frame (the ship or the star). Without knowing who accelerated, you cannot say who is "actually" moving since this would imply an absolute frame (set of spacetime coordinates) against which to measure, and this simply does not exist.
I'm not picking on you, superluminal, but your post is a classic STR-type that I would like to respond to.
The captain KNOWS who accelerated. He turned his engines on and felt the force of the associated acceleration. He feels this force end when he turns his engines off. He knows he has increased HIS velocity relative to the star and the rest of the universe, evidenced by changes in Doppler shifts of recieved light, including the star. The so-called 'absolute' reference frame is simply the reference frame of the star he is calculating his velocity relative to. Special Theory insists the "observer's" frame is the frame at 'absolute rest' and the rest of the universe moves relative to this observer's 'rest' frame.
When there is a relative closing velocity between you and the star, the frequency you measure for the light is increased. This is the same as saying the energy of the light is increased. You measure the same velocity for light in any experiment you an do aboard your ship. The intensity of light is a measure of the luminous flux - basically the number of photons per second - that reach your detector. So as your relative closing velocity increases, the energy of each photon is increased (doppler) and the number of photons per second (flux) increases - the light gets "brighter".
Yes, the frequency of the light is increased because the captain, the observer, IS moving relative to the already emitted light, light that was emitted possibly years earlier, depending on distances. The light is Doppler shifted not because it changed frequency after being emitted, but because the captain, the observer, has increased his velocity relative to the emitted beam.
The light does not change wavelength or speed after being emitted, except for gravitational effects. The change in recorded frequency of the light is due solely to the captain's change in velocity relative to the already emitted beam of light. The speed of light is a constant in co-moving frames. The speed of light is still 'c' when the captain considers his velocity, evidenced by the increased frequency, relative to the emitted beam. The increase in frequency is a measure of a relative velocity speed increase, due to the captain's (observer's) motion WRT the emitted beam.