How to get custom post type categories into permalinks?

I’ve used Pods to create some custom post types, one of which is “Site Story”. Down in Pods Admin > Advanced Options, I’ve given this pod a Custom Rewrite Slug of “story”. Additionally, each Story is assigned to a category: News, Sports, Opinion, etc.

I’d like the permalink for a Story to be something like https://example.com/news/this-is-the-news-story. Unfortunately, I can’t get there: All the permalink settings I’ve tried insist on putting “story” into the url, right after the domain name (https://example.com/story/this-is-the-news-story)., even when the permalink setting is Custom Structure: /%category%/.

Is it possible to get what I want? I’ve experimented with the Custom Post Types Permalinks plugin, but that still wants to lead the permalink with “story”. I’ve seen a few pages that talk about modifying the register_post_type call for the custom post type, but I’m not sure how that fits into the whole Pods thing (if it even does). Is there any advice out there? Thanks!