The Hall Sensors are really just part of a voltage divider setup, Hall Sensor > OpAmp > MCU.
The 4 pins you're talking about using on the Arduino are Analog Inputs, you can't use those to control anything.
You can drive the LSX, LSY, RSX and RSY lines (TP19, 20, 21 and 22) externally with the voltage from a POT, DAC or PWM. They are 0v to 2.8v (with 1.4v being maintained for center) though you may have to remove the OpAmps as they will keep the voltage centered if the Sticks are removed and may have an effect on driving them externally. I've used a PIC/DAC setup before on the RSX line, locked RSY in center with a couple of Resistors, and removed the OpAmp to be safe, so what you're wanting to do can be done.
If you're using one of the newer version DS3 controllers with the SN84001 IC you'll need to scope those 4 lines and make sure they haven't changed the way it works, again, since Sony has nothing better to do with their money. I haven't put the scope on one of those later versions yet to make sure it's the same setup as the ones with dedicated OpAmps, might be, might not be, so check it to be sure. If you don't have a scope handy use a DMM and see what the voltage swing on those TP19~22 lines are while you move the Sticks. If they sit at 1.4v (Sticks centered) and swing around 1v or so in each direction for that line then it's the same setup, if they do anything other than that you'll need to scope them to see what's going on there first.