Use stylesheet from a different location

Create a theme’s style.css as normal but just include the header information. This will be a config file only.

The register and enqueue your stylesheet(s). You can use any path you’d like. I think this is what you need:

function register_theme_styles() {
    wp_register_style( 'theme-styles', site_url( '/css/site.css' ) );
    wp_enqueue_style( 'theme-styles' );
}
add_action( 'wp_enqueue_scripts', 'register_theme_styles' );