Several years ago I was overcharged by the driver of a replacement bus (tram replacement). I knew I was right, but his argument was that because he was working out of his normal area the fare was more than the normal tram ticket (Arriva bus replacing Metrolink).
I argued back that that had nothing to do with the fare, but he wasn't shifting. In the end I paid (put the right change back in pocket and fished out a £20 note in front of him).
I've regretted it ever since. What I should have done was sit on the bus, having had the correct fare refused, and wait for him to either give up and set off, or call the police. By the time they arrived customer services phone line would have been open and would have explained to the driver that he was an idiot (which was the response I got when I rang them later, along with a free ticket).
If the railway makes a mistake, the railway bears the cost, and I wouldn't trust the suggestion of paying and claiming a refund either, once they've got your money in those circumstances they'll be very reluctant to give it back, without proof that it would be impossible for an individual to provide days or weeks after the event.
At least if you challenge it on the train, and there's a group of unrelated people all claiming the same thing, there's a good chance the BTP will realise that something has gone wrong on the railway's side.