I figured it out. While Pressgram (or theoretically any other CPT-creating plugin) can add to a custom post type, I first had to make "live blog entry" visible to other plugins, allowing it to have categories, and also enabling the featured image (thumbnail) for future potential.
This meant changing line 69 to:
'public' => true,
'supports' => array('editor', 'author', 'thumbnail'),
'taxonomies' => array('liveblog', 'category'),
Then, in Pressgram (or other CPT-based plugin), I had to tell it to automatically assign a liveblog to the posts it adds to my CPT. So I found where that plugin creates the posts and added a line like this:
wp_set_post_terms( $post['ID'], '####', 'liveblog');
Where "####" is the ID number of the liveblog page.
It's a hacked together way, but it works for now.