I called HSBC again on Tuesday and, after explaining everything that had happened, was again offered the $15 phone payment option. God give me strength.
I was eventually connected to tech support, and this time the guy fixed the problem in 5 seconds. He was apologetic, and his tone indicated to me that he was the one who typically smoothed over the blunders of his coworkers.
At least I can access my account now. $15 phone payment indeed!