After waiting on hold over 30 minutes on two separate occasions, I called back a third time and this time entered options to redeem for Singapore Airlines (instead of "another airline"). I got connected within a couple minutes, the agent found the United flights I wanted without issue, and I was able to pay using my normal credit card that had failed online and in the app. No charge for booking via phone was asked for or paid, though if they had tried I would've explained the payment problem.

So, happy ending.