I guess you personally don't need assistance anymore but maybe someone else stumbles across this, like I did. I suggest reading the whole thing so you know where this heading and have everything reading once you actually start doing stuff.
Unfortunately, brute force is the only thing that will help here, but it has to be applied precisely. Firstly, the plastic holding the trigger button has to be prevented from budging away (or even snapping) when you apply pressure to the pin, so turn the joystick on its side and put a stack of old wood, or a single piece of wood on top of a stack of books or whatever with a piece of wood on top under the trigger button so that the base is vertical and the stick is horizontal. It doesn't have to be perfectly flat, but it should be pretty close. The pin is a bit shorter then the hole, the side where the pin is slightly depressed should point up. Stick a slightly-thinner-then-the-hole-is-wide nail with the point removed (snap it off with a pair of pliers, or saw it if necessary) into the hole and give it a smack with a hammer. Be careful, but you will need some force. Pick the joystick up and check if the pin moved - ideally your first smack doesn't move it so that you can carefully hike up the pressure you apply. Once it moved a bit you can see it leave a mark on the wood - use that as orientation and make a hole there. I suggest not ******* around at this stage and just using an electric drill to put a 6mm wide, ~1cm deep hole in there. That's far more then necessary, but it's just an old piece of wood anyway. Now place the hole under the pin and give it a few more smacks. Soon you will be able use the part of the pin that sticks out when adjusting the hole's position. Go a bit further, then once there's a good bit of pin sticking out you can grab it with your fingers and try to twist and pull at the same time. The twisting keeps it moving, reducing the friction (exactly what the ABS in your car is trying to prevent). If you can't pull it out just give it a few more smacks with the hammer.
To put it back in just put it back on the improvised stamp and hammer the pin right in there... do make sure that the holes are aligned first.