You're right, it can win but it's unlikely. With full speed for 5 moves (4 turns + initial move) it would move 15.
I moved stuff around while writing this down and I missed the new math. A 5th turn would bring the average speed needed to 2.1, which is much more obtainable. A max speed of 4 would be cool too, but it might be too much in favor of the train.
Train's health too might need adjustments.