 A light ray starts at the origin of the stationary frame K and reaches the location X at time T. During this time the origin of the moving frame advances a distance vT, leaving the distance X-vT for the light to travel to reach X. The light postulate says that the light travels at speed c in both frames, so X=cT, and X-vT=ct. Combining these equations leads to t=T-vX/c0. This is exactly the approximation of the Lorentz transform for v/c<<0. This analysis did not make any use of the clocks of the moving frame.  I don't know why you would claim that. Isn't t the time according to clocks in the moving frame?  So, do you think that he based his light postulate on empirical results? 