Inserting additional AJAX using hook_flag
I'm certain there is a better way to do this, but rather than using hook_flag use JavaScript.

The trick seems to be to use bind('flagGlobalAfterLinkUpdate', function(event, data)

And then catch the specific flag you're using with data.contentId;

(function ($) {
    $(document).bind('flagGlobalAfterLinkUpdate', function(event, data) {
        var fsrnid=data.contentId;
        var fsrpostcard = $(".postcard-" + fsrnid).text();
        var fsrpostcardset = fsrpostcard.indexOf('Yes') &gt;= 0;
        var fsrcalled = $(".flag-customer-called-" + fsrnid).text();
        var fsrcalledset = fsrcalled.indexOf('Yes') &gt;= 0;        
        var fsremail = $(".flag-email-sent-" + fsrnid).text();
        var fsremailset = fsremail.indexOf('Yes') &gt;= 0;

        if (fsrpostcardset || fsrcalledset || fsremailset) {
            $(".completed-" + fsrnid).text('Yes');
        } else {
            $(".completed-" + fsrnid).text('No');
        }

    });
}(jQuery));


The Flag API (JavaScript) documentation is a bit dated and doesn't necessarily apply directly to Drupal 7.  It is still very useful in figuring out how the Flag module uses JavaScript though.