ElementAPI -> Find entry template name
Perhaps instead of trying to get a template name, use the entry type as the template name, similar to the method that you linked to. In the link, it doesn't check which template is assigned to an entry, but instead, uses the entry type to load a template with the same name. In your example that might look like this:

craft()-&gt;templates-&gt;render('path/to/template/' . $entry.type, [
  'entry' =&gt; $entry
])


Where your template would be named the same as the handle for your entry type. Does that make sense?