Dear Georgina,
There is no escape: The coordinate system K extends into the negative direction to the left of A and into the positive one to the right. For any point B located to the left the distance to A becomes closer if it moves to the right, and hence the Doppler effect yields an increased apparent frequency until B moved to the right side of A and the distance between A and B got growing yielding an decreased apparent frequency. Consequently there is no common coordinate system k fixed at B for which all points or in Einstein's terminology clocks read the same apparent time. Calculating 1/(c-v) - 1/(c+v) = 2v /(c^2-v^2) provides a misleading average.
I also disagree with your utterance that there is nothing to add. At least one question is remaining: How sound are Poincaré's Lorentz transformation and its pre-runner by Woldemar Voigt? The latter did deal with an elastic medium instead of light. The mathematical correctness of the belonging derivation of LT has been put in question by Aleksandar Vukelja. Whether Peter Jackson's objection also matters is not yet clear to me. Van Flandern accepted the Lorentz factor as it was used by Lorentz himself in contrast to Poincaré/Einstein.
Having read several related papers including those by Lorentz himself, Fitzgerald, Janssen and Stachel, Harvey Brown, Yakovenko, Ashby, and of course various Wikipedia articles, I see the main case already clarified for good.
Admittedly, I did not yet read the huge heretical literature. Maybe the opponents of Einstein could convince me that he was right.
Regards,
Eckard