How do I render the search form for a particular Apache SOLR search page on its own?
Well I'm not sure what the problem was, but after a bit of further digging into apachesolr_search I've come up with a work around which seems to do the trick:

$search_page = apachesolr_search_page_load('PAGE_ID');

module_load_include('inc', 'apachesolr_search', 'apachesolr_search.pages');
$build['search_form'] = drupal_get_form('apachesolr_search_custom_page_search_form', $search_page);


As kiamlaluno has noted in the comments it doesn't look like it's actually apachesolr_saerch causing this behaviour, but rather the core search module doing...something.

Hopefully this code will save someone the time in the future.