It sounds as if you are doing everything you can to settle out of court, so keep on with that and it may well work.
But you need to know that the railway (edit) may not let you settle out of court: you may end up in court, and be convicted, and face a punishment.
That punishment will be a fine, on top of the figures that the railway has already told you about. But you will not go to prison; quite simply, that doesn't happen for fare dodging.
I see that the charge is under the Regulation of the Railways Act: that means that if convicted then for a year if anyone with a good reason (maybe someone offering you a job) asks you will have to tell them about the conviction. It will also show up on a DBS check, and for roles that need a lot of trust (things like personal care, or where you are handling other people's money) then you may have to mention it for longer than that.
This probably sounds awful, but from what we see here, many employers and so on are prepared to give a second chance, as long as going forward you're honest.
So it's quite likely that after paying your fine, mainly the conviction will be embarrassing. But that's nowhere near as bad as going to prison or not being able to get any job.