Is there a way to switch the Search form (used in the block) to do a Facet Search instead?
As stated below I originally understood your question.  Here is how you change the default search page to the one with faceted navigation:
In admin/config/search/settings you can turn off default node search and make faceted your default as shown in the below screen shot.


The Search API module allows you to choose other search methods, including using something like SOLR server for search, which also does faceted queries.