Creating a tastypie resource for a "singleton" non-model object
This sounds like something completely outside of TastyPie's wheelhouse. Why not have a single view somewhere decorated with @require_GET, if you want to control headers, and return an HttpResponse object with the desired payload as application/json?
The fact that your object is a singleton and all other RESTful interactions with it are prohibited suggests that a REST library is the wrong tool for this job.
I'm using tastypie and I want to create a Resource for a "singleton" non-model object. 
For the purposes of this question, let's assume what I want the URL to represent is some system settings that exist in an ini file.
What this means is that...:

The fields I return for this URL will be custom created for this Resource - there is no model that contains this information.
I want a single URL that will return the data, e.g. a GET request on /api/v1/settings.
The returned data should return in a format that is similar to a details URL - i.e., it should not have meta and objects parts. It should just contain the fields from the settings.
It should not be possible to GET a list of such object nor is it possible to perform POST, DELETE or PUT (this part I know how to do, but I'm adding this here for completeness).
Optional: it should play well with tastypie-swagger for API exploration purposes.

I got this to work, but I think my method is kind of ass-backwards, so I want to know what is the common wisdom here. What I tried so far is to override dehydrate and do all the work there. This requires me to override obj_get but leave it empty (which is kind of ugly) and also to remove the need for id in the details url by overriding override_urls.
Is there a better way of doing this?