First off I will start by saying that I did it on my Vespa LX but ended up replacing the side stand.
Not that I did anything wrong.
My foot loop rusted and eventually snapped from one of the welds. So I took down the side stand and decided to weld a new one. I think it was a pretty decent job at the end of the day although it requires quite a bit of welding skills. Did not find a metal wire at the same diameter so went for a larger one, had a thick metal wire lying around. Made it a bit bigger than the old one and welded it a reinforcement also, although it was purely for esthetic reasons, there's no need for it. Painted the side stand with some black paint in the end (did a lousy job on that one, I agree). On the GTS though it's much more complicated, as it has a very large loop and it's curved if I remember right.
Now to explain why I ended up replacing the side stand. While I was in the process of welding it I stumbled across a friend with a Vespa that had replaced his side stand with an aftermarket one made by Buzetti (around 30EUR in Europe).
2 things I noted from the start.
One, the Buzetti one is more stable, as the foot that stands on the asphalt is designed with the triangle expanding into the outside instead of narrowing to the outside (check picture). I have been using on the old one an extension pad bought from Aliexpress, that, while indeed giving the side stand a lot more stability also lifts a bit the scooter (the pad is machined from aluminum and quite thick, adding around 2-3mm or more to the length of the side stand which makes the scooter stand more upright, which is not necessarily a good thing on an object that has a tendency of falling from the side stand by nature.
Second, the Buzetti one does not fold up the moment you lift a bit the scooter, you have to do it yourself with the foot, just like a bicycle one. I have hated the original one folding up by itself from the first day I got my Vespa, as sometimes you only want to lift it a bit, then put it back on the stand.
Last but not least, the Buzetti one comes in 2 colors, black as the original or chromed for those that want their accessories in chrome.