Am I to understand, then, that I would also increase the water proportionately?
Yup. For caramel, the only purpose of the water is to provide a way to dissolve the sugar. It doesn’t remain in the final caramel, all of it boils off. If you double the sugar then you should also double the water (or add ten percent or whatever adjustment you are making) so that it remains easy to dissolve.
When I make caramel, I put the water in first and then the sugar, and after a moment or two when the water has spread to all of the sugar, I put it on the burner and begin heating. Contrary to what most instructions say, I don’t stir at all, not even once, until the mixture has just begun to boil. This helps keep sugar crystals off the sides of the pan. Once it begins to boil, I remove the pan from the heat and stir slowly. I return the pan to the heat and stir as necessary until all of the sugar crystals have dissolved. Then I continue to heat (without stirring) it until it starts to color, at which point you can swirl it to even out the temperature so that it browns uniformly.
If you were to need to double the recipe, would you do it in two batches instead to avoid this?
No, I have caramelized larger batches of sugar with no problems using my usual method. Is it possible that there wasn’t enough water to dissolve all the sugar? If so, your process may have been similar to making a dry caramel, which definitely works but is harder to control as it browns fast and unevenly. If you’re ever wondering how much water to add, you can just multiply the weight of the sugar by 0.30 to get the weight of the water.