BuddyPress.org

Look at line 1426 of that function:

add_action( 'init', 'bp_core_action_search_site', 5 );

In your bp-custom.php, you can stop that action by doing this:

remove_action( 'init', 'bp_core_action_search_site', 5 );

You’ll need to place that within a function.

See this for more info: https://codex.wordpress.org/Function_Reference/remove_action