I don't agree with those who have said the OP's wife is somehow legally obliged to pay the PF even though it is invalid.
A valid PF- one issued in accordance with the Penalty Fares Regulations- is a valid, enforceable, debt, and must be paid, but an invalid PF is not.
The statute
https://www.legislation.gov.uk/uksi/2018/366/regulation/5 states
Penalty fares
5.—(1) Subject to regulations 6, 7 and 10, if a person fails to produce a platform ticket or a valid travel ticket in accordance with regulation 4, a collector may charge that person a penalty fare."
This particular, specific, PF was invalid because it has no basis in statute, since the OP's wife did in fact produce a valid travel ticket ticket in accordance with regulation 4.
The OP's wife owed nothing when the PF was issued because it was- as it remains now- a legal nullity, and the rejection of the appeals did nothing to change that fact, nor to somehow transform it into an enforceable debt.
The fact that 'the railway' may think invalid PFs are nevertheless legally enforceable is irrelevant; no court would agree. The provision of the Penalty Fares Regulations- namely Regulation 12- providing for civil recovery of unpaid PFs provides for recovery only of those PFs
issued in accordance with 5(1), and this particular specific PF was not issued in accordance with 5(1), because the condition of failing to present a valid travel ticket was not met, since the ticket presented was indeed valid.
If the matter did proceed to a civil court, the court would have one question to decide: whether the PF was issued in accordance with 5(1) and hence is a valid PF; if so, judgment would be entered in favour of the claimant by virtue of Regulation 12; if not, the claim would be dismissed. While it would be wrong to absolutely guarantee a particular decision in the so-called (D)DJ lottery, it is clear that the correct decision would be for the court to dismiss the claim, if brought.
Furthermore, the OP's wife is an a stronger position than most because the clear validity of her ticket means that she has a clear substantive defence to any criminal prosecution, in addition to the prosecution itself being statute barred by any non-perverse interpretation of the Regulations.
In my opinion, there is nothing to negotiate; all the OP's wife needs to do is say that the alleged debt is denied in its entirety, and that she won't be paying a penny until and unless ordered to do so by a court. That said, since courts disapprove of ambush defences, it would also be advisable to state the reasons for the refusal- namely, that the ticket presented was valid; hence the PF was invalid.