Hi David
LS uses SendEmailMessage() internally, the function is found in /application/helpers/common_helper.php. However, you need to load the Yii Framework in order to use that function.
The function then sends an email using the specified configuration from LimeSurvey.
Be also sure to set the parameter $isHtml to true when you like to send formatted mails - otherwise the email body would get flattened to text-only before sending.
Still a simple solution would be to send the message with PHP's mail() function or by using phpmailer (included with LS). In order to have the page not loaded again, you could make the button open the php script in a popup window, and to be nice also display a message that the mail was sent successfully and that the window may be closed now

. If popups are not desired, an AJAX implementation would be required.
Regards,
Gregor