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Photons are emitted by the source in different directions, meaning some photons that do not move straight to the scatterer but a little sideways so that they have a slightly lower x-component, might get shifted to the frequency photons going straight to the scatterer would have without the train. I think this can confuse the measurement but can be solved by making sure only photons in the wanted direction can reach the scatterer.
Also the source might be moving at the right velocity, but I am not sure that alone is enough to let a photon emitted by it to have the same Doppler shift. You can not hold the photon, it is massless, so I am not sure if the movement of the source is enough to influence the photons in the right way after being emitted. There is nothing pulling them ahead after leaving the source.
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