I think there's an easier way to look at it. If the clocks all accelerate in a symmetrical manner in the station frame, we can say that they all print out exactly the same time.To all,
I noticed that no one denies that the printing of the front clock should be ahead of the printing of the rear clock, except that it is hard to define the proper frame for it.
Then it is a simple matter, if one is familiar with the relativity of simultaneity, to say that this directly implies that the clocks are NOT syncrhronized in the train frame.
I'll provide a reference to where Einstein mentioned this. Now, if you don't quite follow Einstein's argument, you would be far from the first. But at least you might have some inkling of where the rest of us are coming from, and what the issue is with your analysis.
The link is https://www.bartleby.com/173/9.html, and the reference is Einstein's book, "Relativity: the special and general theory", chapeter IX, 'The relativity of simultaneity'.
So there you have it. BECAUSE the events on two neary cars are simultaneous in the station frame (when we assume symmetry), it follows, from Einstein's argument, that they can't be simultaneous in the train's frame.Einstein said:
UP to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: 1 Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative.
The "train's frame" is only defined in the limit as the two cars are very close to each other, which has caused some discussion. But after taking said limit, there's nothing at all ambiguous about the train frame - it's moving at some velocity v along the track, and because of the effect Einstein mentioned, the fact that the clocks are synchronized in the station frame implies they cannot be synhronized in the train frame.