Is there an option for radio buttons with Facet API + Search API (database)?
There is a patch for the Facet API Bonus module that adds a radio buttons widget.

The issue with the patch is https://drupal.org/node/2034777

With this patch you also need to use the Facet API module patch mentioned by RajeevK in the comments of your question, which can be found at https://drupal.org/node/1393928

If you try those patches and comment back in those issues with how it worked out for you it will also help progress those issues so that the patches can be committed and you no longer have to use patched modules - a win for all.