How to display a shortcode caption somewhere other than the_content

Try this:

$caption_info = array();

add_filter( 'img_caption_shortcode', 'capture_caption', 10, 3 );

function capture_caption( $blank = '', $attr, $content ) {

    global $caption_info;

    $caption_info[] = array('attr' => $attr, 'content' => $content );
    return ' ';
}

It will save info from all captions into global $caption_info variable and suppress their display in content (space is returned because filter result is ignored if empty).