I'm sorry, but you are incorrect.
Perhaps you would like to explain how 8 years of communications reach the ship as it is 'changing frames' on the star 4 light years away? What you are describing requires information to travel at faster than light speeds. Can you explain when the doppler effect stops working as things get faster? Does the doppler effect stop working at 0.01c? At 0.1c? At 0.5c? Can you provide a source?
Here is what happened to you, perhaps years ago. You read and partly understood the frequently cited explanation about ships passing each other. The situation described in those explanations exists only for a moment as the ships motions are parallel. But as they are approaching each other, their transmissions are dopplered to arrive faster, and as they leave each other, they are dopplered to arrive slower. That's the entire point behind 'redshifting'. By your explanation light in behind and in front would be red shifted (time slowed down). That's what redshifting and blueshifting are... the change in the frequency of light.
And what is frequency? It's how many waves arrive per second. If I transmit 1 million waves of red light, and you receive 1 million waves blueshifted blue light, it takes less time to arrive because the frequency of blue light is higher... if I encode information in those 1 million waves, you receive that information faster when it arrives blueshifted. It doesn't matter if it's blueshifted a little or a lot, or if I'm undergoing relativistic dilation or not... it still arrives faster, and thus the perceived rate of time change is higher.
Also, think about this. The ship is approaching at 0.8c. If there were no relativistic effects then it would perceive the transmissions arriving at 180% normal speed, or 1.8x normal. But because of the time dilation on the ship (60% normal) that 180% becomes 300% (1.8 doppler / 0.6 relativistic = 3.0x perceived) perceived reception rate of the transmission. Similarly as they fly away from each other, without relativity transmissions would arrive at 1/5th speed (0.20 doppler), but because the ship's relativistic dilation is 60%, that 20% becomes 33% (0.2 doppler / 0.6 relativistic = 0.333 perceived).
So relativistic effects occur, but that doesn't mean doppler effects don't.