For anyone viewing this in the future, you just need to add the argument $excluded_terms => 'YourCategoryID' to the_post_navigation, where YourCategoryID is the ID of the category you want to be omitted from pagination.
To more thoroughly explain what I’m looking for, let’s say I have two categories: ‘Blog Post’ and ‘Off Topic’. When users see the “next post” and “previous post” options under the posts categorized as ‘Blog Post’, I don’t want them to see posts categorized as ‘Off Topic’. It’s fine if the Off Topic posts have no navigation under them whatsoever. I just want navigation on the main posts to completely ignore that the ‘Off Topic’ category exists, skipping over to the next ‘Blog Post’ instead.
I have tried that plugin, yes. It works for excluding the category the from home page for example, but it does not fix navigation to stop displaying posts with that category.