Its not clear what the letter you have received is actually saying. It puzzles me that you refer to it as a Summons Application and at the same time a demand for payment of a Fine. It doesn't seem right to me that both would be the purpose of the same Letter.
However, the situation you describe appears less surprising to me than it does to you.
You are perhaps right to describe buying the wrong ticket as 'stupid', however, the real Offence occurred when you chose to travel on a journey for which the ticket didn't authorise you to travel. Your mistake should have been corrected, you shouldn't have travelled to Barking without a ticket for that journey - that is 'stupid' bit, and now you have apparently an opportunity to pay some sort of charge to make amends.
I suggest you pay it as soon as possible, because what you have done is to commit a Criminal Offence, and (without knowing what terms the letter offers you, I can't be clear) you are presumably the subject of an Investigation into that Offence with a view to gathering Evidence to support a Prosecution in Court.
I think you'll agree that any payment is better than a Prosecution!
Hopefully you'll be more careful in future to travel on the journey that your ticket authorises you to take, and not another journey for which you haven't paid. I guess the letter also warned you of the maximum Fine that a Court can impose for such an Offence? That might also remind you of the importance of having the correct ticket for your future journeys.
Please do let us know if the letter offers any other route to a resolution of the matter, and, purely out of curiosity, what amount is the "bigger fine" the Company has proposed?
[I expect somebody else will be along shortly to tell you that only a Court can impose a Fine. But I guess you are just using that word colloquially and that the letter does not imply that you have been Fined by the Company].