The posts are important as unpublished? Or with a future publish date?

I understand the issue. The Search & Filter Pro cache is not seeing these post as having been changed and thus does not make changes to the cache for them. as soon as you edit the post and hit update, without making any changes, the posts are flagged as having been changed and thus the cache knows to update itself for each post.

I have not done this for some time, but last time I had to do something like this I used a plugin:

https://wordpress.org/plugins/custom-bulkquick-edit/

I selected all the posts, made no changes, and hit the update button. Maybe that would work for you?

Are you able to give me admin logins to a live site so that I can take a more detailed look into this for you? You can reply here, but use the private reply option at the bottom of the post.

You will need to trigger our cache as each post inserts, using an action from WP All Import. This code would go into your child theme functions.php (if you haven’t made a child theme, you need to follow the guidance for the theme you have, or the generic guidance from the WordPress Codex and/or the many Internet guides on it):

add_action('pmxi_saved_post', 'post_saved', 10, 1);

function post_saved($id) {
  do_action("search_filter_update_post_cache", $id);
}

Sadly, any that are already imported cannot be caught by this.