Send Processing Order Email from custom payment plugin

You can do it very easily by using his mail class. you can extend that class as well. Here is the code which should work with your class code. if it did not work than you need to extend the woocommerce email class same like you did for your custom payment gateway.

 $mailer = WC()->mailer();
        $mails = $mailer->get_emails();
        if ( ! empty( $mails ) ) {                
            foreach ( $mails as $mail ) {
                if ( $mail->id == 'new_order' || $mail->id == 'customer_processing_order' ){
                    $mail->trigger( $order->id );                    
                }                 
            }            
        }