As has been commented before, GD doesnt do a very good translation to 2-colours, especially for photos.  The following routine converts to two colours, I believe using error diffusion (the algorithm's nicked off news).  It's slow, but just about adequate for small images and low load.  I suspect it can be made much more efficient :-)
function ImageColorFloydSteinberg($dst_img, $src_img) {
    ImageColorAllocate($dst_img, 0,0,0);
    ImageColorAllocate($dst_img, 255,255,255);
    $grey_img = ImageCreate(ImageSX($src_img), ImageSY($src_img));
    for ($a = 0; $a <= 255; $a++) ImageColorAllocate($grey_img, $a,$a,$a);
    for($x = 0; $x <= ImageSX($src_img); $x++) {
        for($y = 0; $y <= ImageSY($src_img); $y++) {
            $color = ImageColorsForIndex($src_img, ImageColorAt($src_img, $x, $y));
            $greyscale = .299 * $color["red"] + .587 * $color["green"] + .114 * $color["blue"];
            ImageSetPixel($grey_img, $x, $y, ImageColorClosest($grey_img, $greyscale, $greyscale, $greyscale));
        }
    }
    for($x = 0; $x <= ImageSX($src_img); $x++) {
        for($y = 0; $y <= ImageSY($src_img); $y++) {
            $color = ImageColorsForIndex($grey_img, ImageColorAt($grey_img, $x, $y));
            if ($color["red"] > 128) {
                ImageSetPixel($dst_img, $x, $y, ImageColorClosest($dst_img,255,255,255));
                $err = $color["red"] - 255;
            } else {
                ImageSetPixel($dst_img, $x, $y, ImageColorClosest($dst_img,0,0,0));
                $err = $color["red"];
            }
            if ($x != ImageSx($src_img)) {
                $color2 = ImageColorsForIndex($grey_img, ImageColorAt($grey_img, $x+1, $y));
                $newgrey = $color2["red"] + $err * 7 / 16;
                ImageSetPixel($grey_img, $x+1, $y, ImageColorClosest($grey_img,$newgrey, $newgrey, $newgrey));
            }
            if ($x != 0) {
                $color2 = ImageColorsForIndex($grey_img, ImageColorAt($grey_img, $x-1, $y));
                $newgrey = $color2["red"] + $err * 3 / 16;
                ImageSetPixel($grey_img, $x-1, $y, ImageColorClosest($grey_img,$newgrey, $newgrey, $newgrey));
            }
            if ($y != ImageSy($src_img)) {
                $color2 = ImageColorsForIndex($grey_img, ImageColorAt($grey_img, $x, $y+1));
                $newgrey = $color2["red"] + $err * 5 / 16;
                ImageSetPixel($grey_img, $x, $y+1, ImageColorClosest($grey_img,$newgrey, $newgrey, $newgrey));
            }
            if ($x != ImageSx($src_img) && $y != ImageSy($src_img)) {
                $color2 = ImageColorsForIndex($grey_img, ImageColorAt($grey_img, $x+1, $y+1));
                $newgrey = $color2["red"] + $err / 16;
                ImageSetPixel($grey_img, $x+1, $y+1, ImageColorClosest($grey_img,$newgrey, $newgrey, $newgrey));
            }
            
        }
    }
    imagedestroy($grey_img);
}
To output your WBMP, use
ImageWBMP($final_img, "", ImageColorClosest(255,255,255));