I feel your pain! I have had a slightly different issue with Iberia seating assignments over the past few days which took multiple phone calls and eventually the use of Japan Airlines' and Royal Jordanian's websites to resolve!
Iberia's website, software and phone agents are horrible. On one call I got the same comment as you "[1/13, 12:13 PM] Iberia: Even though they are Iberia flights, the ticket belongs to another airline, which does not allow us to assign the seats." which is a blatant lie as it is the operating airline that almost always handles seat assignments.

In your case, I have no idea why the online seat selection and card payment would fail. My only suggestion would be to keep calling and try to get a competent agent who will do your seat allocation over the phone and process your credit card payment at the same time. You may need to call multiple times.
Were you calling the US number: 800-772-4642 or the Spanish number: +34 913 33 67 01? I have had better luck with the Spanish number, and there is an English language option.