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But the anti-aliasing causes pixels on the boundary of the black outline and the colored interior to be darker versions of the interior color, by mixing in the black. So simply looking for an exact match of the original color fails to replace the boundary pixels.

I already took that into consideration.

This problem can be solved, though: you could test every pixel for the value of the brightest RGB component, and deduce from that by which factor it is darkened compared to the interior color. And then darken the replacement color by the same factor.

Here's what I'm doing. I collect all the color indexes. For each color index with a non-zero alpha value, I make a determination of whether it belongs to the outline of a piece or to its coloring. I do that with this conditional:

if (color_diff(array($cv["red"], $cv["green"], $cv["blue"]), $original) 
<= color_diff(array($cv["red"], $cv["green"], $cv["blue"]), "black")*2) {

The color_diff function returns the greatest difference between one color's minimum component value and the other one's maximum component value. I added the *2 to prevent some errors, and that part now seems to be working well. If it belongs to its coloring, I recolor each RGB component with this function:

function recolor_component_as ($originalvalue, $currentvalue, $newvalue) {
    if ($originalvalue == 0)
        return $newvalue;
    return min(255,round($newvalue*($currentvalue/$originalvalue)));
}

One remaining problem is that when I try to color the Black pieces white, I get an aquamarine fringe around the Black Motif pieces. In this case, $originalvalue will be 255 for the red component and 0 for the green and blue components, and $newvalue will be 255 for all three components. So, it recolors some edge pixels with positive values for green and blue but with 0 for red, resulting in aquamarine. So, I might have to factor in the value of each component when deciding how to recolor each component. But when I tried that earlier, I got some undesirable results.