parent-theme-mod-loader.php

<?php
/**
 * Plugin Name: parent-theme-mods-loader
 */

add_action(
	'customize_register',
	/**
	 * Hooks in customize_register to load our control, register settings and do what needs to be done.
	 *
	 * @param WP_Customize_Manager $wp_customize The WordPress Customizer object.
	 * @return void
	 */
	function( $wp_customize ) {
		/**
		 * A custom control.
		 * Contains a simple button to allow loading a set of options.
		 *
		 * TODO: This class should be on a separate file.
		 */
		class WPTRT_Parent_Theme_Mods_Loader_Button extends \WP_Customize_Control {

			/**
			 * The control type.
			 *
			 * @access public
			 * @var string
			 */
			public $type = 'wptrt-load-settings-buttons';

			/**
			 * An array of options that will be passed-on to the JS.
			 *
			 * @access public
			 * @var array
			 */
			public $choices;

			/**
			 * Enqueue scripts.
			 *
			 * @access public
			 * @return void
			 */
			public function enqueue() {
				// TODO: Enqueue script (currently added via an anonymous function in customize_controls_print_footer_scripts).
			}

			/**
			 * Pass variables to our JS template.
			 *
			 * @access public
			 * @return void
			 */
			public function to_json() {
				parent::to_json();
				$this->json['choices'] = $this->choices;
			}

			/**
			 * JS template for the control.
			 *
			 * @access public
			 * @return void
			 */
			protected function content_template() {
				?>
				<label>
					<span class="customize-control-title">{{{ data.label }}}</span>
					<# if ( data.description ) { #>
						<span class="description customize-control-description">{{{ data.description }}}</span>
					<# } #>
				</label>
				<button class="button">{{ data.choices.button_text }}</button>
				<?php
			}
		}

		// Register control-type. Whitelists the control for JS templating.
		$wp_customize->register_control_type( 'WPTRT_Parent_Theme_Mods_Loader_Button' );

		// Get theme details.
		$theme = wp_get_theme();

		// Get the template name (parent theme).
		$template = $theme->get( 'Template' );

		// If we found a parent theme, add control.
		if ( $template ) {

			// Add control.
			$wp_customize->add_control(
				new \WPTRT_Parent_Theme_Mods_Loader_Button(
					$wp_customize,
					'wptrt_load_parent_theme_mods',
					[
						'section'  => 'title_tagline',
						'settings' => [],
						'priority' => 999,
						'choices'  => [
							'setting'     => 'theme_mods_' . $template,
							'button_text' => esc_html__( 'Load parent theme settings' ),
						],
					]
				)
			);
		}
	}
);

add_action(
	'customize_controls_print_footer_scripts',
	/**
	 * Add footer scripts.
	 *
	 * @return void
	 */
	function() {
		/**
		 * TODO: This should be on a separate file that gets enqueued
		 * in the WPTRT_Parent_Theme_Mods_Loader_Button::enqueue() method.
		 */
		?>
		<script>
		wp.customize.controlConstructor['wptrt-load-settings-buttons'] = wp.customize.Control.extend( {
			/**
			 * Runs when the control is ready.
			 */
			ready: function() {
				var control = this,
					button  = control.container.find( 'button' );

				// When we click on the button trigger the loadSettings method.
				button.on( 'click', function( event ) {
					event.preventDefault();
					control.loadSettings();
				});
			},

			/**
			 * Load settings and replace existing values in controls.
			 */
			loadSettings: function() {

				// Data for the AJAX request.
				let data = {
						action: 'wptrt_load_parent_theme_mods',
						nonce: '<?php echo esc_html( wp_create_nonce( 'wptrt_load_parent_theme_mods' ) ); ?>',
						setting: this.params.choices.setting
					};

				// Make the call.
				jQuery.post( ajaxurl, data, function( response ) {
					let settings = [];
					// Try to decode the response.
					try {
						settings = JSON.parse( response );
					} catch (error) {
						// Nothing to do.
					}

					// Make sure the response is an object now.
					if ( _.isObject( settings ) ) {
						
						// Loop settings.
						_.each( settings, function( val, key ) {
							
							// Find the control.
							let control = wp.customize.control( key );

							// If the control exists, change its value.
							if ( control ) {
								control.setting.set( val );
							}
						} );
					}
				} );
			}
		} );
		</script>
		<?php
	},
	999
);

add_action(
	'wp_ajax_wptrt_load_parent_theme_mods',
	/**
	 * AJAX Callback.
	 */
	function() {

		// Security: Check if the nonce is valid.
		check_ajax_referer( 'wptrt_load_parent_theme_mods', 'nonce' );

		// Sanity check: Make sure we've got everything we need.
		if ( isset( $_POST['action'] ) && 'wptrt_load_parent_theme_mods' === $_POST['action'] && isset( $_POST['setting'] ) ) {
			
			// Get the setting-name.
			$setting = sanitize_text_field( wp_unslash( $_POST['setting'] ) );

			// Print the option value.
			echo wp_json_encode( get_option( $setting, [] ) );
		}
		wp_die();
	}
);

I had a thought.
theme_mods is a sparse array. I wonder if it would be desired to fill the holes with the parent theme's default values. Doing that is a little trickier.