I'm making progress with this. If I understand correctly, I need to use a Toolset archive with a View rather than the default WordPress archive to be able to do this, yes?
I made a new Toolset archive and loaded the WPBakery template I used for my CPT as the loop template, and that works okay. If I then change the archive purpose to custom search, and add a search filter with a dropdown and Ajax search settings, the WPBakery shortcodes don't work, which is a pity, but not the end of the world.
Is it possible to have a dropdown in a widget which does the same thing?