Unfortunately GWR effectively have you 'bang to rights' here. They can prosecute you for travelling without a valid ticket. It doesn't matter that you didn't realise your Railcard was expired; the fact that it was expired meant it was no longer valid and in turn, your ticket was not valid. As such, I would suggest that accepting a settlement with GWR is likely to be cheaper than being prosecuted and convicted, which is the likely alternative.
Does the letter have any attachments or anything else that would explain how they have arrived at their figure? I can only presume that you were you travelling on a return ticket, as an Anytime Return routed "Any Permitted" from Leamington Spa to London Terminals is £178. That plus an assumed £110 for their costs would equal £288. But that would not match their stated calculation method, of charging the
difference to this fare - and you do state that you held an (otherwise) valid ticket.
In any case, since you travelled to Paddington, the appropriate undiscounted fare would be the £143.50 Anytime Return routed "via Banbury". Charging the difference to that would already shave a fair bit off the cost. I think it would be reasonable to politely ask GWR to consider reducing their settlement cost to the difference from the fare you actually paid, to £143.50. Based on their stated methodology you would hope they would agree to this.
Condition 9 of the National Rail Conditions of Travel does state that an operator can recover "its costs of this follow up (including claim or prosecution costs) from you ... costs claimed will not exceed the actual costs of such follow up steps taken", though the degree to which this is enforceable is questionable. I think it is worth considering whether paying the settlement now to avoid losing the opportunity of settling before Court, but subsequently challenging GWR as to how £110 (a conveniently round figure) represents their "actual costs" given that this would appear to be a very simple case.
