Comparing the Doppler predictions for a moving body assuming (1) lightspeed fixed in the observer's frame, (3) lightspeed fixed in the emitter's frame, and (2), special relativity, we find that special relativity's predictions are the geoemtric mean, or root product average of the two earlier predictions.
|-||fixed observer||Special relativity||fixed emitter|
|recession Doppler||1 : 1/(c+v)||1 : SQRT( (c-v) / (c+v) )||1 : (c-v)/c|
|transverse Doppler||1:1 (no effect)||1 : ( 1-vv/cc)^0.5||1 : 1 - vv/cc|
|Lorentzlike factor exponent||0||0.5||1|
This allows us to associate SR's numerical predictions by assuming that lightspeed is globally fixed in the emitter's frame or the observer's frame and still obtainn the same physical prediction.
So, for instance, if we assumed that the speed of light was fixed wrt the observer, then light from a body receding at v=0.8c would be shifted by the ratio 1/1.8= 9/5 by the propagation shift, and then also by the Lorentz redshift of SQRT((1-0.8)/1+0.8)) = SQRT(0.2/1.8) = SQRT(1/9) = 1/3 . The final frequency seen would then be the original frequency emitted, multiplied by 9/5, and then by a third, giving a total motion shift of 9/5 × 1/3 = 3/5 = 0.6.
On the other hand, if we believed that the speed of light was globally fixed wrt the emitter, we woudl expect the motion shift component to be 1-0.8 = 0.2, and we'd expect our own measurements ot be using clocks that ticked too slowly by a rate of 3:1 due to our own time dilation due to motion, giving a final result of 1/5 × 3 = 3/5 = 0.6
If we now take the case of transverse-moving body, if we assume that the speed of light is globally fixed in the observer's frame, then we expect no propagation-based shift effect, but only see the Lorentz redshift due to SR time dilation of 1/3, so that the frequency that we see is one third of that emitted.
Assuming instead that the speed of light is globally fixed for the emitter, we'd calculate an expected aberration redshift of 1/9, but also expect our reading to be greater than this by a factor of three, due to the time dilation of our instruments, giving a total visible shift of 3/9 = 1/3.
The purpose of the Lorentz transform under special relativity is to generate a single set fo equations that can be interpreted by any inertial observer as being caused by a fixed speed of light in any frame, combined with the time-dilation of any bodies that ar emoving with respect to that arbitrarily chosen frame. No matter which frame is selected, the results are the same.
This if often described by saying that SR says that the speed of light is fixed for the observer, it might be more accurate to say that, if the SR equations are correct, we cannot prove that the speed of light is not fixed for the observer, but we also cannot prove that it's not fixed for any other simple inertial frame. While the speed of light is taken as globally fixed, the frame for which is is globally fixed is an "unknowable".