How does one do that when standing at the airport counter with two screaming kids and a flight leaving in an hour?
Please read the OP. He has asked what he can do to avoid this. The answer is to deal with UA in advance, not at the counter with "screaming kids."
I still suggest that there must have been something odd about the purchase of the first domestic ticket by OP. Not wrong or bad, just odd. Perhaps he had to reenter the CC information multiple times, entered the wrong CVV more than once or some other odd situation which tripped the fraud alert.
Or it was a glitch and will never repeat itself.