Filebeat only has one output, if you send your data to one Elasticsearch cluster you cannot send it anywhere else.
The easiest solution would be to send your data to a Logstash server and this Logstash server send the data to your current Elasticsearch cluster and your third-party, but it seems that you can not do that.
The other way would be to query your Elasticsearch cluster using Logstash and send the data to your third-party, but this would not be in real time and can be expensive and add some load to your Elasticsearch cluster depending on the volume of data.