Harvey Brown and Oliver Pooley believe that length contraction is more fundamental than the constancy of the speed of light. First "a certain motion-induced deformation of rigid bodies" takes place and then, as a consequence, "light is measured to have the same speed in each inertial frame":
Harvey Brown: "The FitzGerald-Lorentz (FL) hypothesis was of course the result of a somewhat desperate attempt to reconcile the null result of the 1887 Michelson-Morley (MM) experiment with the hitherto successful Fresnel-Lorentz theory of a stationary luminiferous ether, a medium through which the earth is assumed to move with unappreciable drag. The MM experiment is rightly regarded today as one of the turning points in physics, and although it is discussed widely in textbooks, it is remarkable how much confusion still surrounds its structure and meaning. In order then to understand the FL hypothesis, it is necessary first to go over some welltrodden ground; sections 2 and 3 below are designed to show what the 1887 null result does and does not imply. In particular it is shown in section 3 that IN THE CONTEXT OF A THEORY OF LIGHT IN WHICH THE LIGHT-SPEED IS INDEPENDENT OF THE SPEED OF THE SOURCE, A CERTAIN MOTION-INDUCED DEFORMATION OF RIGID BODIES, OF WHICH CONTRACTION IS A SPECIAL CASE, IS REQUIRED."
Harvey R. Brown and Oliver Pooley: "One then appeals to the relativity principle again - the principle entails that these coordinated contractions and dilations must be exactly the same function of velocity for each inertial frame, along with the principle of spatial isotropy, in order to narrow down the deformations to just those encoded in the Lorentz transformations. What has been shown is that rods and clocks must behave in quite particular ways in order for the two postulates to be true together. But this hardly amounts to an explanation of such behaviour. Rather things go the other way around. It is because rods and clocks behave as they do, in a way that is consistent with the relativity principle, that light is measured to have the same speed in each inertial frame."
Harvey Brown, Oliver Pooley,
When the observer starts moving towards the light source, wavecrests start hitting him more frequently (Doppler effect). Reasonable scientists conclude (see below) that the speed of the wavecrests relative to the observer has increased, that is, the speed of light as measured by the observer varies with the speed of the observer. Can you refute reasonable scientists' conclusion by referring to some crucial "motion-induced deformation of rigid bodies"?
"vO is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + vO. (...) The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time."
"La variation de la fréquence observée lorsqu'il y a mouvement relatif entre la source et l'observateur est appelée effet Doppler. (...) 6. Source immobile - Observateur en mouvement: La distance entre les crêtes, la longueur d'onde lambda ne change pas. Mais la vitesse des crêtes par rapport à l'observateur change !"
Carl Mungan: "Consider the case where the observer moves toward the source. In this case, the observer is rushing head-long into the wavefronts... (...) In fact, the wave speed is simply increased by the observer speed, as we can see by jumping into the observer's frame of reference."
Roger Barlow, Professor of Particle Physics: "Moving Observer. Now suppose the source is fixed but the observer is moving towards the source, with speed v. In time t, ct/(lambda) waves pass a fixed point. A moving point adds another vt/(lambda). So f'=(c+v)/(lambda)."
Tony Harker, University College London: "If the observer moves with a speed Vo away from the source (...), then in a time t the number of waves which reach the observer are those in a distance (c-Vo)t, so the number of waves observed is (c-Vo)t/lambda, giving an observed frequency f'=f((c-Vo)/c) when the observer is moving away from the source at a speed Vo."
Albert Einstein Institute: "As the receiver moves towards each pulse, the time until pulse and receiver meet up is shortened. In this particular animation, which has the receiver moving towards the source at one third the speed of the pulses themselves, four pulses are received in the time it takes the source to emit three pulses [that is, the speed of light as measured by the receiver is (4/3)c]."