Figured it out. It seems that there are additional parameters on the SwaggerResponse attributes that I was not aware of (due to a lack of any clear documentation on the Swashbuckle package). I found these by using the ObjectBrowser and found that there is an additional set of optional parameters:
[SwaggerResponse(HttpStatusCode.OK, Description = "Get a Company by ID", Type = typeof(DataContract.Company))]
Anyways, with that in place, the REST client is generated correctly. I hope this helps anyone trying this themselves...