Update to this thread - here's what worked and didn't for me:

Chase Sapphire Preferred - failed, bank security couldn't see the transaction (helpful agent)

Barclaycard US Airways World Master Card - failed, bank security could see the transaction but it still wouldn't go through once security settings were changed (very helpful agent)

Citibank-issued AA Amex - failed, didn't call

Gold Amex - failed. Then read that Amex's tend not to work.

Then, I followed the additional advice on Seat61 and registered for an account on Renfe. Into that account I saved a Bank of America Visa Signature... which worked on the first try!

So, some combination of having a Renfe account and using a Bank of America card worked. Too bad B of A will charge me a 3% foreign transaction fee, but, that's better than $30+ for using Petrabax (or $100+ for Rail Europe since they don't offer the discount fares).