If profileImage() seems to be doing nothing — and "inverted colors" during a CMYK > RGB conversion is a sign of this — check that ImageMagick has the lcms delegate available.
From a command prompt:
convert -list configure | grep DELEGATES

If you don't see lcms in the list then Imagick won't do any color profile conversions, and won't give any warnings about this.  In that case, install the Little CMS library ( http://www.littlecms.com/ ) and recompile ImageMagick.

If you need to convert images that are on CMYK format into RGB and want to preserve colour information, this may be helpful:

<?php
$image = new Imagick("CMYK_image.jpg"); // load image
$profiles = $image->getImageProfiles('*', false); // get profiles
$has_icc_profile = (array_search('icc', $profiles) !== false); // we're interested if ICC profile(s) exist

if ($has_icc_profile === false)
{
  // image does not have CMYK ICC profile, we add one
  $icc_cmyk = file_get_contents('/path/to/icc/SomeCMYKProfile.icc');
  $image->profileImage('icc', $icc_cmyk);
}

// Then we need to add RGB profile
$icc_rgb = file_get_contents('/path/to/icc/SomeRGBProfile.icc');
$image->profileImage('icc', $icc_rgb);

$image->setImageColorSpace(Imagick::COLORSPACE_RGB);

$image->writeImage("RGB_image.jpg");

?>

There may be better and more elegant ways to do this, but hope this helped.

Instantiating an image files, and attempting to save them in PNG format using setImageFormat('PNG'), I occasionally encountered
"gd-png: libpng warning: iCCP: profile 'icc': 'CMYK': invalid ICC profile color space".

However, transformimagecolorspace( \Imagick::COLORSPACE_SRGB ) had not effect, because getImageColorspace() did not indicate COLORSPACE_CMYK.

Instead, one has to check for the presence of any ICC profile,  inspect the ICC profile header (offset 16) for the "CMYK" color space tag, and then remove the ICC profile:

$ICC_count = remove_ICC( $image, 'CMYK');

function remove_ICC( \Imagick $image, ?string $colorspace ): int
{
    $icc_count = 0;
    
    foreach ( $image->getImageProfiles( 'icc' ) as $name => $profile ) {
        if ( empty( $colorspace ) or 0 === strcasecmp( $colorspace, substr( $profile, 16, 4 ) ) )
            // Either consider all ICC profiles, or just those matching the color space.
            $icc_count++;
    }
    
    if ( $icc_count > 0 )
        $image->profileImage( 'icc', null );        // Delete any ICC profiles.
        
        return $icc_count;
}

Thanks for this very valuable information. 
For a further push in the correct direction please 
consider that you have to download the profiles, 

and your most likely sources are:

http://www.color.org/srgbprofiles.xalter

and

http://www.adobe.com/support/downloads/product.jsp?product=62&platform=Windows

Note that the profiles are free, but you must install them 
to make them available on you host system.