Sorry to intervene but you’re trying to use 2 filtering plugins into one.
Search&Filter is a tool to filter the posts as WPSP is too (but you filter them within the admin page).
What you are trying to achieve here is to use the design of WPSP with search filter. For this, you need to customize the output of Search&Filter with the CSS of WPSP. I don’t see how you can use both any other way they both run 2 different queries.