Does Total Cache have an API?

Hello Shannon,

Thank you for reaching out and I am happy to assist you with this.
Short answer -yes. In wp-content/plugins/w3-total-cache/w3-total-cache-api.php you can find the purge functions.
In your case:

function w3tc_flush_post( $post_id, $extras = null ) {
	$o = \W3TC\Dispatcher::component( 'CacheFlush' );
	$o->flush_post( $post_id, $extras );

or you can call flush for the URL:

function w3tc_flush_url( $url, $extras = null ) {
	$o = \W3TC\Dispatcher::component( 'CacheFlush' );
	$o->flush_url( $url, $extras );

As you can see you can call w3tc_flush_post( $post_id ) or w3tc_flush_url($url)
I hope this helps!
Thanks!