According to chat support, the 24 hour lockout only extends from your most recent attempt:

You: I tried probably 30 times. Does that mean I'll be locked out for 720 hours?
Agent Jordan R: No. Just 24 hours. Do not attempt again, as you will only reset the 24 hour timer if you do.
Also, their billing system is run by rogue robot chimpanzees from outer space:

You: I tried with Firefox and Internet Explorer, as well as Chrome. None of them work. Is there any way you can reset my timer? There's no reason to be locked out for 24 hours after your processor declined my card.
Agent Jordan R: Unfortunately, the billing system is a fully-automated system. There is no person or persons overseeing the transactions and manually choosing which cards to decline. But once an error 401/406/601 is given, a 24 hour hold is placed on the account and any further attempts for payment will automatically get declined by the billing system. At this point, we can only advise to wait 24 FULL hours and try again tomorrow, preferably using a different browser and/or PC. Unfortunately, there is nobody in the Support Center with manual controls to the automated billing system and we do not possess the proper tools to bypass these errors. We sincerely apologize for the frustration this may be causing you, but unfortunately, we can only provide the suggestion to please remain patient and try again in 24 hours.
I guess I'll be trying again tomorrow and hoping real hard that it doesn't happen again, because that would mean another 24-hour lockout that apparently Square Enix itself is unable to bend - just in time for the weekend yay! I never imagined it could be so difficult to spend money.