This would require an inmense amount of work, not worth the hassle for a cheap device.
Limited angle? Then adjust sensitivity in game. Maybe your issue comes elsewhere, like dead angle, input filtering, all that needs checking first before blamign the potentiometer.
Every console has used potentiometers for the analog sticks, and there's no such delay.
Probably this device is like this, cheap and limited, with slow sampling rates, so no amount of work will fix it
...
Not pretending to be a jerk, but next will be: Help me to make a circuit, help to to program it, help me to calibrate it, help me to fix broken pcb... Please do for me!
Quickly you'll realize it gets too complicated and drop the idea!
It might be $300, but you get something that works properly out of the box (Or just get a used one).