Amazon UK refund or open A-Z

I used 2nd class parcel service and item says deleivered with gps but buyer is insistant it hasnt been.

Ive done the usual check neighbours/will report stolen threats. But my question is do i just refund, offer to send replacement or ask them to open a a-z case where i will still have to refund but will need to appeal the defect to get it removed.

could making them go through a-z wean out any chancers or will it just make more effort for me?