So it turns out that it's as easy as sending a click even to the button. While this is probably more of a work around than an actual solution, it does get the job done in a single piece of code.

    jQuery(document).on('click', '.social_score_wrapper', function() {
        jQuery('my-button-selector').click();
    });

So it turns out that it's as easy as sending a click even to the button. While this is probably more of a work around than an actual solution, it does get the job done in a single piece of code.

    jQuery(document).on('click', '.my-element', function() {
        jQuery('my-button-selector').click();
    });