It's a long shot but if OP bought oil and filters at O'Reilly or Auto Zone or Advance, and if OP has the rewards card at any of those stores and if OP showed his card each time he bought oil and filters, then the store manager just might be able to pull up his purchase history and print that out for OP to show the dealership he really did buy oil and filters and change the oil. OP will need his record of the dates and mileage of each oil change, and it would help back up his case if he has proof of all other maintenance done as well. That scenario is a very long shot though.
OP could also ask the dealer to pull off the valve cover, or possibly also pull the oil pan if they have not done so already. A good dealership will agree to that and will check for sludge or other evidence of lack of maintenance/oil changes. OP would have to agree to pay the labor time for removing the pan and cover if the warranty claim winds up denied. A good tech or service manager can tell if an engine has not been taken care of. If the valvetrain looks good, they could possibly use that as verification the oil has been changed regularly and then a good dealership will use that inspection as proof enough OP or someone else has been changing the oil on schedule. The key to all that is, whether the dealer is willing to check the valvetrain and go to bat for OP with the warranty administrator.
This also depends heavily on how the extended warranty is written. If it specifically states OP has to provide all sales receipts and records of all maintenance, whether it was DIY or not, OP and the dealer may not get much help from the warranty admins. I really hope the dealer and warranty admins are able to do the right thing and help OP with this problem.