Hide Customizer’s Widget Choices From The List

The only way to disable a Widget from appearing in this list is to unregister it. You can do that just in the customizer:

function unregister_customizer_widgets() {
    if ( is_customize_preview() ){
        unregister_widget('WP_Widget_Recent_Posts');
    }
}

add_action( 'widgets_init', 'unregister_customizer_widgets' );

But that will remove it from the page preview and the sidebars. Even if it is used somewhere on the page it will not be displayed, that is not a very useful solution. You could check if the widget has been used by looking in the sidebars_widgets option and hide it only if it had not been used, but again it is not very good solution.

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Your best option here is to hide it with CSS and enqueue stylesheet only for the customizer:

function my_customize_preview_style() {
    wp_enqueue_style( 'new_customizer_style', get_template_directory_uri() . '/customizer-style.css' );
}
add_action( 'customize_controls_print_styles', 'my_customize_preview_style' );

Inside it hide the desired widgets and remember about !important as they all have inline display: block;.

But consider again if you have to hide the widget from the user, displaying in in one place and hiding in the other will be confusing.