Trouble with Peltiers TEC and USB Output Current
You bought an element rated for 5 V 20 W. This means that it will draw ~4 A when connected to 5 V stable supply. Period. It won't draw more. It won't draw less. There is nothing you can do about this, it is practically a ~1.2 Ohms resistor. The Peltier is not "defaulting" to anything because it has no idea that someone has some limit.

To get your Peltier battery  to run at 2.4 A from 5 V, you need to get an element rated for 5 V 12 W. Or get the element as suggested in the video, 12V-60W, which comes down to resistance of 2.4 Ohm.

And yes, the charger likely shuts down at 3.9 A.