TLDR: Use "Request Desktop Site" option in your mobile browser. Had to post here as this thread helped me think through the problem and work around it to complete the payment. Was still getting the same problem on iPad and iPhone IOS8 being asked to append the security key to my password when there was no button to request one. I'm using the Chrome IOS browser but you can probably do the same with Safari. When you get asked to append the security key tap the options icon (a vertical set of 3 dots on Chrome) and choose "Request Desktop Site". This then reloaded the page, displaying the button to request the SMS, which promptly arrived and allowed me to log in and complete the payment. Spent a couple of hours of my life faffing around trying to solve this one! Grrrr!
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