Afope, is it clear from this image why the company is not allowed to prosecute you?
https://www.railforums.co.uk/threads/wmc-summons-i-bought-a-ticket.195403/#post-4295828
They aren't allowed to put someone through the hassle of the penalty fare system and then have a second go by prosecuting.
If you're uncertain, you can simply write to the company saying politely that you don't understand why they are proposing to prosecute when the regulations "appear to me to say" that they are not allowed to.
If you like, a student union might be able to provide advice, or alternatively some solicitors (look for one dealing with crime) offer a free initial consultation.
As they can't prosecute, you don't need to offer a settlement (and you aren't guilty of getting on the train without a valid ticket in any case, though it might have been hard to persuade the magistrates.).
Their other mistake was to ask you to pay too much for the penalty fare. This is a general issue that all the companies got wrong.
https://www.londontravelwatch.org.uk/news/view?id=777&x[0]=news/list
Paragraph 16 (3) (a) of the Penalty Fare Regulations (first link above) says that if the penalty fare is not imposed in accordance with the Regulations (which includes it being for the wrong amount) that's grounds for appeal. So if you'd known about that mistake by the company, you wouldn't have had to pay anything (again, they aren't allowed to hassle you too much).
So you could point that out to the company as well.
You can pay the £39.80, which would have been the correct penalty fare, or you could argue that you aren't liable because it wasn't imposed in accordance with the Regulations and the Regulations say it would be grounds for appeal. Here again, you could write to them that "my understanding is" that it's for the wrong amount because it isn't for the "fare applicable" as defined under the Regulations, rather than saying whether you are refusing to pay.
The good news is that they can't prosecute you at all for this. If you chose not to pay the penalty fare, the very worst possibility would be that they could pursue a civil claim. That's very different from being able to prosecute for a criminal offence.