The key to this is your statement:
"I get 14.4 volts at the battery terminal until the car warms up. As it warms the volts at the battery terminal climb to 16 volts or more..."
The difference between the alternator running in your vehicle and being tested on the bench is heat. I suspect the voltage regulator's reference climbs when it gets hot, and so the regulated voltage climbs with it. This happens in the vehicle as the vehicle warms up, but not on the bench, where the test is short and there is no external heat applied to it.
I suggest taking it back to where you had it tested and have them replace the voltage regulator with a new one. It's the least work, the least cost, and the most likely solution to the problem.