Whether or not this would be allowed would generally call for a more fact rich situation than the one presented in the original question, that would cast light upon why a retailer might be inclined to refuse to accept payment. Hypothetical legal questions that presume that people are acting irrationally for no good reason are generally ill posed and don't have meaningful answers.
Sometimes, there might be a legitimate privacy interest implicated if the invoicing party acknowledged a payment from a third party. For example, suppose that the invoice was for a paternity test and payment would confirm that the incapacitated person actually obtained a paternity test.
Sometimes, there are legal rights beyond payment that are implicated and the reasonableness of a refusal might hinge on those rights. For example, suppose that the invoice was for an option to keep using an oil well. Payment of the invoice by the deadline would keep the oil well operating and the land owner sending the invoice might prefer that it not be paid so that the oil well would be shut down. Quite a few contracts are structured in this way. Maybe the invoice was for the right to purchase a first edition of a book when it was finally released, for example, and not paying it would free up a copy for someone else at a price that had increased in the meantime.
Or, suppose that the invoice were for unpaid taxes and payment of the taxes would prevent property from being seized for sale by tax authorities, and the taxing authority would prefer that the invoice was not paid so that the valuable property could be liquidated.
But, it is hard to imagine that there would be any reason that an ordinary retailer with an ordinary bill would ever refuse payment, although I suppose that this might trigger an interest or penalty amount owed under the contract for late payment. If that were the case, the principal of mitigation of damages, which says that a party to a contract must take all reasonable steps to mitigate their damages, might obligate the retailers to accept the payment or forfeit the penalty amounts that the retailer could have avoided as damages by accepting payment.
I doubt that an undelivered tender of payment from a third party would eliminate the obligation, but, it might limit the damages that could be claimed as in the scenario above.