Have you checked whether the key is still under warranty.
Call the service department at your local $tealer, talk to a service advisor NOT the service co-ordinator who will try to schedule a visit.
Explain to the SA that you bought key just under a year ago and your hassles so far with the key FOB. Ask if they have a test/equipment that they run to check the key.
or
Call around and check if a locksmith has a IR & RF key fob tester.
That should eliminate the key FOB

If it is not the key FOB, you going to need to get a diagnostic done on the central locking system.
So you may need to go to the dealer/indy or a friend with a GT1.

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