Can I get the relationships of a node using node_load()?
As an option you can index the entire rendered HTML of your content. 
Let's assume your index is called "default_node_index" and content type is "article".


Visit /admin/config/search/search_api/index/default_node_index/workflow
Enable "Complete entity view"

Visit /admin/config/search/search_api/index/default_node_index/fields
Enable "Entity HTML output".

Manage display of your content type /admin/structure/types/manage/article/display
Enable "Search Index" in "Custom display settings" fieldset.

Visit /admin/structure/types/manage/article/display/search_index
Add all the fields you want to render. You can extend this display using the regulat hooks, templates and preprocesses if needed.