I am not a legal profsessional and ths post is not legal advice
My understanding of Penalty Fares is that you are only obliged to pay the actual fare on the spot (First class Anytime Single between the stations he travelled), and the remainder of the Penalty Fare can be paid at a later date. As we don't know which stations he travelled between, I don't know if the £7 he had on him would have covered the on-the-spot payment
Penalty Fares Rules section 8 states
Strategic Rail Authority said:
8.2 The authorised collector may require that person to make a
minimum payment that is equal to the full single fare which
they would have had to pay for their journey if penalty fares
had not applied. The person must be given 21 days to pay the
rest of the penalty fare, starting on the day that penalty fare is
charged. If the penalty fare is not paid in full immediately, it
must be paid in any way that is set out in the notice referred to
in rule 8.3
However, there is no right to a Penalty Fare, and the Train Operating Company can bring a criminal prosecution instead under the
Railway Byelaws, section 19. The good news is that that a prosecution under the Byelaws is not recordable so won't show up in most subsequent searches. It only usually results in a fine.
(There's also the more serious Regulation of Railways Act which a criminal prosecution could be brought under, but as the Train Operating Company would have to prove
intent which is a higher burden of proof, I doubt it would be used in this case)
Again you've not been very helpful in not actually telling us what this 'slip of paper' says so it's hard to give advice (I am willing to bet the term 'season
al ticket' doesn't appear anywhere on it), however if the case has been referred to the prosecutions department then I would advise contacting writing to their prosecutions department with an explanation of what happened, and seeking an out of court settlement with them.
As you have now learnt, on the railways, the law is stacked against you, so next time, tell the pushchair users to go in First class instead,