Publish button inside custom field group

Try this:

<div class="my-metabox">
    <!-- This is your metabox HTML -->

    <-- Add this button somewhere: -->
    <button id="my-submit" class="button button-primary button-large">Submit</button>
</div>

Add this JavaScript snippet however you want only to this screen (either by checking the post type and enqueue this in the admin footer or output it directly along with your metabox HTML):

<script type="text/javascript">
    jQuery(function($) {
        // You can substitute this with a CSS rule somewhere:
        // <style type="text/css"> #submitdiv { display: none } </style>
        // if you wish to avoid the hiding 'flicker'
        $('#submitdiv').hide();

        $('#my-submit').on('click', function(ev) {
            ev.preventDefault();                
            $('#publish').trigger('click');
        });
    });
</script>