You can avoid storing the sent messages on the phone by setting the SMS Gateway as default. Just open the navigation drawer on the app and click on "Make Default" as highlighted in the below screenshot.
Making the app default messaging app will prevent the storage of both sent and received messages so make sure you have the "Read Received" option turned on so you can see the received messages in the SMS Gateway app.
If you won't revert the above changes then you can open your primary messaging app and it will prompt you to make it a default app. Just choose "Yes" and you will be able to store sent and received messages in your primary messaging app.