So, the timing here is absolutely crucial.
At the FIRST point in time that you were asked to present your ticket, had you already completed the purchase of it?
If you had, then you should persist with your appeal on those grounds.
However, the letter does state otherwise. Is it wrong?
- What time did you board the train?
- What time did you complete the ticket purchase?
- What time were you asked for your ticket?
- What time was printed on the Penalty Fare that was issued?
Note that there is a byelaw that requires you to buy your ticket before you board when you can do so. Buying it after the train has departed does not make the ticket invalid, but neither does it retrospectively cancel out the offence of boarding without a valid ticket. A Penalty Fare focusses on whether you are in possession of a valid ticket at the point in time when you are asked to show one. (Unless there are exceptional circumstances, you can only face one sanction, not both. Once they've decided to use a Penalty Fare and especially after an appeal has been decided, they can't change theirs minds and prosecute.)