re: I want, or rather I do, use Global Buddypress Search as a plugin.
It is not possible to integrate with the core WordPress search because it is highly optimized for only searching WP pages/posts. The Global Buddypress Search plugin effectively replaces the core WP search in order to search its custom data structures. Depending on how the plugin is developed it might actually be possible to incorporate additional custom searches but I’d have to delve deep into their code to know. Even if does it would require a lot of custom coding in order to do so.
re: since Connections Pro is not using Custom Taxonomy, the plugin finds no content for Connections Pro, which actually is pretty uncool.
Connections does not use Custom Post Types (CPT) for its data because the db schema is CPT’s are optimized for post like data. Connections uses a db schema more optimized for directory content. I think you’ll find that much larger plugins such as WooCommerce and Easy Digital Downloads are even moving away from using CPT’s because using them are too inefficient for the custom data schema’s they require.
I apologize, I know this is not the answer you want to hear.