Different logic in Facets functionality
In the end I did all of this with jQuery alone. 

I created a block that contains the 


All (23)
Alpha (10)
Bravo (4)
Charlie (0)
Delta (1)
Echo (8)


options, I add the number with jQuery by counting the grouped results, and by JQuery again, I hide the appropriate grouping blocks. So the Facet API block was of no use to me and neither was the View's exposed filter.