According to my analysis of a moving partially reflecting surface, I noticed that the transmitted field should be partially dragged by the moving object. I think that the same phenomenon exists in the Fizeau's experiment with the superposition of two waves having different phase velocities and different amplitudes.
To isolate experimentally the field with higher velocity (c/n + v), I propose to use a signal processing.
https://www.researchgate.net/publication/310753426_Proposition_for_electromagnetic_analysis_of_moving_media_and_physical_explanation_of_Fizeau%27s_experiment
Article Proposition for electromagnetic analysis of moving media
Principles of relativity for SR already are in agreement with Lorentz Transformations and don't need for LT to be introduced again by some other way.
Frizeau did the experiment in 1851. Lorentz published transformations in 1887 to 1892, but they were not recognized as explanations of the 1851 experiment until 1905 with publication of SR. Relativistic addition of velocities in SR is derived from nothing more than constant speed of light, without reference to LT.
So I guess the answer is yes you don't need the LT, but no you do need constant light speed, and the proof is not difficult.
One can consider that the luminiferous medium within the moving column of water is being at least partially dragged along with the flow. No need for Lorentz transformations.
Lorentz gave the explanation of contraction of the distances between the charges based on solution of the wave equation for the potentials. One can read it in the dissertation of Janssen (Ch. 3):
www.mpiwg-berlin.mpg.de/litserv/diss/janssen.../IntroI.pdf
So Lorentz explained the results of all experiments with the moving bodies (or the EM phenomena in the moving medium). Actually, Lorentz didn't need in the postulate of constancy of the speed of light in any frame.
Lorentz used a "local time" variable which is obtained by assuming implicitly the constancy of the speed of light in any frame. This the same variable with the same implicit assumption which are used by Voigt in 1887.
Lorentz:
https://en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies/Section_V
Voigt:
https://en.wikisource.org/wiki/Translation:On_the_Principle_of_Doppler
I wish to say that most authors on the courses of the classical electrodynamics note that Lorentz didn't consider his transformations of the coordinates as reflection of transformation of the space-time. In Sec. II of his paper:
https://en.wikisource.org/wiki/Translation:Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies/Section_II
Lorentz uses the Galilean transformations of the coordinates in order to obtain the direct solution of the wave equation.
Moreover, Lorentz derived its contraction hypothesis from Eq. (23) of Sec. II in a very elegant way.
He reasonably concluded that if a single charge has a shape of the ball (the surface of the charge must be equipotential surface), the surface of the moving charge should be the equipotential too. Because the equipotential surfaces of his Eq. (23) are the contracted ellipsoids, the shape of the moving charge should be the contracted ellipsoid too.
Why is it the hypothesis? Because Lorentz in 1895 couldn't know what is the elementary charge.
So Lorentz used his transformations as an auxiliary tool to solve the wave equations but without physical meaning. Moreover Lorentz suggested that if the dominant factor in creation of the molecules is the EM forces, then in the moving bodies the distances between molecules should change (his contraction hypothesis).
If Lorentz knew the structure of the solid state (cristalline lattice) he would can justify his hypothesis. At least, to explain the results of Trouton-Noble and Michelson-Morley experiments, one doesn't need in transformation of the time variable because the arms of the devices contract in such a way that the measurements give the null result.
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