Ok I guess the issue is this line number 315 in the file framework/presentation/custom-post-types.php –
'rewrite' => array( 'slug' => 'faculty' ),
which lets the staff profiles to have the faculty as the URL prefix as in
https://www.livemeshthemes.com/invent/faculty/judith-powell/
instead of default staff prefix. WordPress is intercepting your URL and messing it up pagination leading to 404 error.
My suggestion – can you change the page URL to something like http://keithschool.birdinhandmedia.com/our-faculty/ so that you can still use faculty prefix for individual faculty members?
Another alternative is to change the line number 314 to
'has_archive' => true,
and let WordPress generate faculty page automatically for you. But then you lose some of the flexibilities of a page with custom content in it.