First without a valid Retail receipt showing you as the original owner will be difficult to get Warranty services unless they don't require a receipt and just goes by your GPU or Pre-built PC Serial Number.
Second since the GPU came preinstalled on a Retailed PC you need to go through the Manufacturer of your PC Warranty.
All hardware inside a Pre-built PC is covered by the Manufacturer of the PC. Most likely you have a one year Warranty depending on the manufacturer.
You can open a Warranty ticket with the PC Manufacturer and see what they say.
If you live in the EU you have automatically a 2 year Warranty from the Retailer that they must comply with under EU Consumer laws.
That shop doesn't seem to be very reputable. I would check to see what Government Consumer department there is in your area that applies to Retail Shops and file a complaint if they refuse to help you with your Warranty claim.