[zwave_proxy] Send HomeID upon client connect (#11037)

esphome/components/api/api_connection.cpp

@@ -1389,6 +1389,11 @@ void APIConnection::complete_authentication_() {
     this->send_time_request();
   }
 #endif
+#ifdef USE_ZWAVE_PROXY
+  if (zwave_proxy::global_zwave_proxy != nullptr) {
+    zwave_proxy::global_zwave_proxy->api_connection_authenticated(this);
+  }
+#endif
 }
 
 bool APIConnection::send_hello_response(const HelloRequest &msg) {