I was trying to make things easy for you. But... if you insist on accelerating the train, just be sure to do it right. Accelerate each piece of the train uniformly (according to observers on the train) so that each piece is always moving together according to observers on the train. Do this right and the train will be accelerated and the clocks will not be affected (as far as folks on the train can tell).
That's not quite correct; the observers on the train will still observe the front clock running fast and the back clock running slow. Heuristically, recall that in an accelerated frame, clocks in the direction of acceleration run fast, and clocks on the other side run slow.
As a concrete example, consider this case.
The (x,t) coordinates of the left edge of the train is given by:
x = 1s * c * cosh (tau / 1s)
t = 1s * sinh (tau / 1s)
And the right edge is given by
x = 2s * c * cosh (tau / 2s)
t = 2s * sinh (tau / 2s)
You can check that:
tau is the proper time along these worldlines.
The lines of simultaneity for the observers at each end are lines that pass through the origin.
The train always has length (1s * c) according to the observers. (as measured along a line of simultaneity)
The right clock always reads twice the left clock. (again, according to the lines of simultaneity)
In general, analyzing from a stationary frame, if the train is accelerated towards the right, and you choose two pairs of events that train observers think occur at the same time, then the left edge will have a greater space displacement and lesser time displacement than the right edge; the proper time experienced by the left edge cannot be as much as that of the right edge.