How to make a stand-alone button to the post edit view

First of all you need prevent whole post form sending.
Try:

function rt_custom_button(){
    $html="<div id="major-publishing-actions" style="overflow:hidden">";
    $html .= '<div id="publishing-action">';
    $html .= '<button onclick="send_note(event);" accesskey="p" tabindex="5" class="button-primary" id="custom" name="">send media note</button>';
    $html .= '</div>';
    $html .= '</div>';
    echo $html;
 }
 add_action( 'post_submitbox_misc_actions', 'rt_custom_button' );

Then you need to add JS function somewhere:

function send_note(event){
    event.preventDefault();

    $.post(ajax_url, {
        action: 'send_note'
    }, function(response){
       ...
    });

}