How to get bounds with react-leaflet
Try this.refs.map.leafletElement.getBounds.
According to the documentation:
You can directly access the Leaflet element created by a component using this.leafletElement in this component. This leaflet element is usually created in componentWillMount(), except for the Map component where it can only be created after the container is rendered.
which is a round about way of saying they store the leaflet object as leafletElement property on their component objects.