I called them to get this offer on my Card - you can do this even if your card isn't directly eligible.
Terms and Conditions
Offer valid at American Express accepting locations only. Prior to refueling your vehicle, you must ensure that the location accepts American Express.
Offer is limited to $40 credit per Card and only spend on this Card counts towards the Offer.
Additional Cards are not eligible for this offer.
Excludes transactions where you do not spend directly with your Card to which the offer is saved, in-person at BP. Valid at New Zealand locations only.
Excludes transactions made through a third party establishment or payment processor.
Credit is not redeemable for cash or other payment form.
Credit should appear on your billing statement within 5 business days from qualifying spend but may take up to 90 days from the offer end date.
Credit will not be applied to your Card Account if your Card has been suspended or cancelled.
Credit may be reversed if your qualifying purchase is refunded.
This is what I got from their Live Chat: The offers are based on the spend pattern of the card and they can be enrolled only if it shows in the Amex offer section in the main home page. Sometimes there may be offers that are available for you and they may be unavailable for others. Also there may be times. you may receive the offer after few days as they are added in batches.