Actually, there is a thing called payment pre-image. Once the invoice is paid, the payment pre-image is given to the payer. Any node involved in the payment routing also knows it. It can be used to prove that the invoice has been paid.
Is this pre-image in some way linked to the initial payment request of the merchant? I mean, if the pre-image only proves that I payed something but not the initial payment request of the merchant then this is not a good prove. If the initial payment request of the merchant and the pre-image are linked together then the same aproach with signed payment requests can be used.