I have had a solution through customer support - here are the emails they have sent me:
Total 3 emails:
My card now works again and I am now playing the game

I hope this helps others!
Edit: (On the second email they did ask me to hide the long card number but only showing the last 4 digits and the name on card - obviously for security reasons).
Edit: I AM NOW BACK ONLINE AND PLAYING!
First email:
Dear Customer,
Thank you for contacting the Square Enix Support Centre.
Following upi from our recent email conversation regarding the error 406 you reported,
We would like to ask you to please ensure that there are enough funds in the account the card belongs to and please send us an image of the card to
sel-id@square-enix.com to be able to whitelist it and for you to use it.
We apologise for any inconvenience caused.
Kind regards,
The Square Enix Support Centre team
Thank you again for contacting us and for your interest in Square Enix products!
More help and information is available at the Square Enix Support Centre website.
http://support.eu.square-enix.com/
Second email:
Dear (my name),
Thank you for contacting the Square Enix Support Centre.
We thank you for the picture.
We have forwarded this.
You will receive an email as soon as we have an update for you.
We would once again like to apologise for any inconvenience caused.
Kind regards,
The Square Enix Support Centre team
Thank you again for contacting us and for your interest in Square Enix products!
More help and information is available at the Square Enix Support Centre website.
http://support.eu.square-enix.com/
Final email:
Dear (my name),
Thank you for contacting the Square Enix Support Centre.
We are contacting you regarding your issues when trying to register a payment method.
The issue with your card should now have been resolved. Please try registering the card again and let us know whether it is working.
We apologise for the inconvenience. Should you have any further issues, please do not hesitate to contact us.
Kind regards,
The Square Enix Support Centre team
Thank you again for contacting us and for your interest in Square Enix products!
More help and information is available at the Square Enix Support Centre website.
http://support.eu.square-enix.com/