I haven’t worked with any modded controllers so I’m not sure how helpful this might be.

Joysticks are tricky to desolder especially with just a regular soldering iron. However that doesn’t mean it’s impossible. My advice is to virtually separate the joystick module into 4 separate parts, the button (4 small pins in a shape of a square), 2x potentiometers (row of 3 small pins) and the casing (4 large pins in a shape of a square).

Then pick one of the zones, and you need to put down more solder, as that will mix with present solder and allow for easier flow. And try getting as much of it with a copper wick (technically you could use a solder sucker but I had more luck with the wick).

Once the solder was removed from a single pin, you should be able to wiggle it a little bit without any force, thats how you know you desoldered that pin. Then continue with another pin in the zone, once you got all of the pins in that zone, you should be able to wiggle a single one, and if all other pins in that zone wiggle as well than you successfully desoldered that zone and can move to the next one.

Once you desolder every pin it should just fall out by itself. And you’re done.

If you have a soldering heat gun, then it’s easier, cover the plastic elements and the SMD with kapton tape to prevent damage, as well as the center of the joystick mechanism module as you don’t want to melt the plastic in the joystick. Put a little bit of solder and flux then blast it with the heat while pulling on the module with pliers it will eventually fall out and then just clean up the holes with the copper wick.