Need advice on theme customizer and child themes

I would comment if able.
If your theme variations consist solely of style modifications you can add the variation stylesheets @ inc/css/ and then here is an example of the references to make it all tick.

@ \options\options-core.php:

$fields_list[]  = array(
    'name'      => __('Select a Skin Color', 'mytheme'),
    'id'        => 'mytheme_skin_color',
    'std'       => 'default',
    'type'      => 'images',
    'options'   => apply_filters( 'mytheme_skin_color', array(
        'default'   => $directory_uri . '/inc/css/skins/images/default.png'
    ) )

@ \options\options-customizer.php:

/********** Class for skin color selection option starts *************/

class mytheme_skin_selector extends WP_Customize_Control {
    public $type="radio";
        
            images-skin-subcontainer {
                display: inline;
            }
            .images-skin-subcontainer img {
                margin-top: 5px;
            }
            .images-skin-subcontainer img.of-radio-img-selected {
                border: 5px solid #5DA7F2;
            }
            .images-skin-subcontainer img:hover {
                cursor: pointer;

                //if get mytheme mod background image has a value then we need to set default bg to none
                $test_skin = $this->value();
                $name="_customize-radio-" . $this->id;
                $selected = ( $test_skin == $value ) ? 'of-radio-img-selected' : '';
                ?>
                
                                         
                        " name="" link(); checked( $test_skin, $value  style="display:none;" />
                        " class="of-radio-img-img " /
            }
 /********** Class for skin color selection option ends *************/

 // mytheme skin 
$wp_customize->add_setting( 'mytheme_options[mytheme_skin_color]', array(
    'default'        => 'default',
    $wp_customize   -> add_control( new mytheme_skin_selector( $wp_customize, 'skin_color', array(
        'label'     =>  __( 'Skin Color', 'mytheme' ),
        'section'   => 'mytheme_design_section',
        'settings'  => 'mytheme_options[mytheme_skin_color]',
        'choices'   => apply_filters( 'mytheme_skin_color', '' ),
    ) ) );

Hope this helps either way.

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