Getting past the painting puzzle
Turns out that there's a slight offset bug involved, but that it only affects where the blue circles appear, and not the accuracy of the game recording your click.
So the solution is to click in the centers of the above circles (or hitboxes) despite the fact that the circles may appear to be incorrect.
(The offset bug works like this: Under Proton I'm forced to run in windowed mode to be able to play the game; wherever the game window itself is on the screen is how much the circles are offset by. So even if I position the window as close to the upper-left corner as I can, my window manager disallows the titlebar being offscreen vertically, and so all my circles are off by (0,22) or however tall my titlebar is. If my window is, say, centered on the screen, the circles are way off.)