Think of it this way: you've learned one of life's important lessons relatively cheap. £256 is not a great deal of money in the great scheme of things, and for that, you've learned how expensive it can be to fail to 'play by the rules'. It was you who let a £25 fumble spiral out of control. Don't let it happen again.
Now, if you could buy a decent credit rating for £256, would you do it? Unless you have no interest in how such things work, or a strong urge to self-destruct, of course you would. You'd be mad not to. Especially if you are, as you clearly are, the kind of person who likes to live now and pay in installments. So that's your starting position: be willing to take it on the chin, take the 'life-lesson' on board, and move on.
You might do better. As others have said, they're institutionalised robbers, but they're built on self-interest, just like you, and they're often open to 'a deal'. Go and see them, bite your tongue, keep your cool, and make your apologies, however much it galls you. Be contrite - it could save you a packet! Show that you recognise you got it wrong but won't ever do it again and is there anything they might be able to do...especially as you're looking to buy this great new bike and every little helps. (Which starts them thinking 'new bike? Maybe he'll want a loan. I could arrange that and get some brownie points. He sounds like he means it and he won't screw up again...').
They may say no, in which case, I say again, bite your tongue, pay up and move on. But I wouldn't be at all surprised if they say yes. Either way, you're on the way to repairing your credit rating, cheaply. But do focus on that: forget the bikes, forget the injustice, forget everything but the importance to you, being who you are and how you like to live, of a good credit rating. Think of them as a bastard sergeant - you can hate them all you like, but you know if you fight them they're going to win...
As for the bike, I'm with others. I've had a good job for 30 years, I've cycled every day, I've taken part in Sportives and the Etape du Tour and I've never even thought about spending that kind of cash on a bike. But each to their own. You say you can afford it, and it is your money.
But do get your ducks in a row: credit rating first, bike second.
