The header and footer templates are applying to SearchWP’s search results page, but not the page template. SearchWP’s query string parameters prevent Bricks from loading a page template at the very least. How do I apply a Bricks page or search results template to SearchWP’s search results page?
Never used SearchWP, but I’ll chime in just as a fill in answer until someone else comes along.
If the page/search template doesn’t work, templates might not be compatible (yet) with SearchWP.
What about your STG page? It looks like that page is loading the search results from the URL variables. Maybe editing that page will get you the results you need?
As an aside, what makes you choose SearchWP over the search functionality Bricks already provides?