Inject class in body when particular page template is used

there is a filter for that and it’s called… body_class 😉
This should work for your case:

function add_my_custom_body_class( $classes ) {
    if ( is_page_template( 'page-customer.php' ) ) {
        $classes[] = 'body_customer';
    }

    return $classes;
}
add_filter( 'body_class', 'add_my_custom_body_class', 10, 1 );

if you wanna add class based on page ID (as you’ve mentioned in comments), you can change the logic inside the function and add class only when particular page is queried, let’s say your page ID is 3:

if ( 3 === get_the_ID() ){
     $classes[] = 'your_additional_class';
}