How can I retrieve indexed facet options for a particular field using Search API?
I seem to have found the answer after some major trial and error, and some digging through the FacetAPI source code. The issue is linked to the same issue explained in the post 
Display facets on non-search pages

I created a facet block in Views for each facet that I'm trying to access, and set them to "hidden" in the block settings within Views. Then by adding a block from my custom module directly below them in the same region, I was able to access the facet data using the following code:

$searcher = 'search_api@MY_CUSTOM_SEARCHAPI_INDEX';
$realm_name = 'block';

$build = facetapi_build_realm($searcher, $realm_name);
krumo($build);


The krumo output contains a render array for each facet you're trying to access, and all the facet options are available for you to access.

I hope this helps someone struggling with the same issue.