One interesting thing about this scenario is that if the rod is straight before it is released in the train frame, then it won't be straight while it is falling, and vice-versa.
The effect is very small for a short rod in normal gravity, but it's important if you want to properly consider what happens to the rod when the wires are cut, particularly in the platform reference frame.
Let's say the rod is straight while it is falling. This means that when it is hanging from the wires, it must be slightly curved. The centre of the rod hangs lower than the ends. The ends of the rod pull upward on the centre to support it against gravity.
When the wires are cut, the centre of the rod doesn't begin to fall immediately. It is still supported by the stress in the rod.
The ends of the rod fall faster than g at first because the stress in the rod is pulling them downward. This is somewhat like the
falling slinky effect, something everyone should try for themselves some time.
So in the platform frame, when one wire is cut first the rod doesn't simply swing like a hinge from the other end... there is a wave of stress-release that travels along the rod.
But, I still think that the bigger issue is that the shape of the rod is relativistically distorted, and that the effect of forces on the rod on the rod's proper shape need to be considered in the rod's rest frame.