The Dev Tools Console application is for Elasticsearch API's. We do have an API for interacting with these objects documented here. We have an issue open to add support to Console to also support Kibana API's here.
I am not sure what exactly is needed other than setting savedSearchRefName as modifying these documents directly is not something we provide support or documentation for. So, you're free to experiement but it would be a "proceed at your own risk" sort of thing.
As an aside, we are working on a new visualization called Lens currently in Beta that is much more flexible. I am not sure about changing a search source, but you can change between visualizations types.