Just as a side-note (IANAL), in the EU, retailers can't refuse to fix something just because it's past their warranty period. If you can establish that the product has broke under reasonable use then you can argue that the product wasn't fit for purpose when you bought it, and therefore it is still the retailer's responsibility to repair it, replace it or give you a partial refund. I did this with Apple when my logic…
The arguing, was it face-to-face or via email. I can do email, but I hate arguing with people face-to-face. Also, if it was face-to-face, was it with just one person, or did it take escalating (and if it did, how many levels did you need to escalate, and who made the final decision?). Sorry for all these questions, I just want to become a more informed and effective consumer.
They offered to fix it for roughly £400, and I:
* quoted the specific law, and had a print out from the relevant website - it was either Which or Gov.uk or something reputable (i.e. not one of my Hacker News comments)
* demonstrated that others had the same problem - I found posts on StackOverflow and the Apple forums
* that I had used the laptop reasonably - it wasn't damaged in any way, and I had a list of the things for which I used the laptop
* I also pointed out that I was a great Apple customer - I had an iPad, an iPhone, and spent a fair bit in the App Store.
The rep went to his manager, and came out a few minutes later saying they agreed to fix the issue for free.
I'm quite introverted and hate confrontation as well, so my advice would be to pre-plan but without making specific scenarios of what might happen. If you have things in writing, it makes you more confident and it makes the reps a bit more nervous as you have a record of the conversation. Also, definitely keep things impersonal. Make it clear that you're not attacking the rep, or even the company, but just the specific policy which you disagree with. Finally, don't make threats like "I'll post this on social media" or "I'll tell all my friends not to buy from you" as the rep's response will typically be "Go on then". Keep calm, professional and friendly, and a majority of the time the reps will be reasonable and try to help you.
Hope that helps!