I have noticed that there is no props for the StatusBar component (React Native) to include an icon or logo of any kind as in this documentation: https://facebook.github.io/rea
For iOS it's not possible to add any custom icons in the status bar, apple doesn't allow it.
For Android you should use Notification and Notification Manager. Just write a native module in java who do this when your app is launching. React Native allow you to call Native Module written in Java, take a look here https://facebook.github.io/react-native/docs/native-modules-android.html
You should also read this for Notification How to show an icon in the status bar when application is running, including in the background?