Consider replacing current approach with this plugin (still need to remove h1, blockquote, etc from formats): Add custom styles in your posts and pages content using TinyMCE WYSIWYG editor. The plugin adds a Styles dropdown menu in the visual post editor.
/*
Plugin Name: Custom Styles
Plugin URI: http://www.speckygeek.com
Description: Add custom styles in your posts and pages content using TinyMCE WYSIWYG editor. The plugin adds a Styles dropdown menu in the visual post editor.
Based on TinyMCE Kit plug-in for WordPress
http://plugins.svn.wordpress.org/tinymce-advanced/branches/tinymce-kit/tinymce-kit.php
*/
/**
* Apply styles to the visual editor
*/
add_filter('mce_css', 'tuts_mcekit_editor_style');
function tuts_mcekit_editor_style($url) {
if ( !empty($url) )
$url .= ',';
// Retrieves the plugin directory URL
// Change the path here if using different directories
$url .= trailingslashit( plugin_dir_url(__FILE__) ) . '/editor-styles.css';
return $url;
}
/**
* Add "Styles" drop-down
*/
add_filter( 'mce_buttons_2', 'tuts_mce_editor_buttons' );
function tuts_mce_editor_buttons( $buttons ) {
array_unshift( $buttons, 'styleselect' );
return $buttons;
}
/**
* Add styles/classes to the "Styles" drop-down
*/
add_filter( 'tiny_mce_before_init', 'tuts_mce_before_init' );
function tuts_mce_before_init( $settings ) {
$style_formats = array(
array(
'title' => 'Download Link',
'selector' => 'a',
'classes' => 'download'
),
array(
'title' => 'Testimonial',
'selector' => 'p',
'classes' => 'testimonial',
),
array(
'title' => 'Warning Box',
'block' => 'div',
'classes' => 'warning box',
'wrapper' => true
),
array(
'title' => 'Red Uppercase Text',
'inline' => 'span',
'styles' => array(
'color' => '#ff0000',
'fontWeight' => 'bold',
'textTransform' => 'uppercase'
)
)
);
$settings['style_formats'] = json_encode( $style_formats );
return $settings;
}
/* Learn TinyMCE style format options at https://www.tiny.cloud/docs/configure/ */
/*
* Add custom stylesheet to the website front-end with hook 'wp_enqueue_scripts'
*/
add_action('wp_enqueue_scripts', 'tuts_mcekit_editor_enqueue');
/*
* Enqueue stylesheet, if it exists.
*/
function tuts_mcekit_editor_enqueue() {
$StyleUrl = plugin_dir_url(__FILE__).'editor-styles.css'; // Customstyle.css is relative to the current file
wp_enqueue_style( 'myCustomStyles', $StyleUrl );
} 
