A retailer is obliged to remedy a problem (by repair or replacement) that arises due to an 'inherent fault' (which was there at the time of purchase) for 6 years after the date of purchase. For the first 6 months of that period there's an automatic assumption (unless there's clear evidence to the contrary) that any problem must be due to an inherent fault. Thereafter it's up to the customer to show that an inherent fault is the cause.
So the Co-op seem to be working within the law. However, although many websites state that the customer must 'prove' that the problem has come about through the presence of an inherent fault, I can see no reason why the normal assumptions of civil law shouldn't apply, meaning that you'd only have to 'show' (based upon the balance of probabilities) that the problem was due to an inherent fault. So simply challenging the Co-op to suggest any reasonable alternative cause might be enough to secure your rights. (i.e. a bit of hassling might be worthwhile!).
Section 75 of the Consumer Credit Act makes Barclaycard jointly liable with the Co-op for ensuring that your legal rights are met but that may well not help you as Barclaycard could also point out, that because it's over 6 months from the date of purchase, it's up to you to prove/show that an inherent fault lies behind the problem.
The 12 month guarantee from Beko is irrelevant here, since it's entirely separate from your statutory rights with the retailer (and card company). However the threat of bad publicity can often change the attitude of a manufacturer, so hassling them again (with mention of 'going to the press') might still be worth a try.