I got the same problem, so what I did is to enter the 'second' surname as 'NOTENGO' which works, then email their support team to remove it and see if it works. Reason for doing this is I need at least an Iberia member number before I can email their support team.
Even I managed to successfully remove the second surname, I am still sceptical their system works because I can login, but if I click on anything with the word 'PROFILE', it will kick me out. Their system looks very fragile.
I have a feeling Iberia support is non-existent. Tried their email, facebook, twitter and you might get somebody understand what you said, but most of the time they dont seems to understand English at all. Never come across any airline support that is so frustrated !!