Neurologists can diagnose autoimmune connective tissue diseases but rheumatologists are internists who are trained to diagnose and treat them and are usually a better choice. It can depend on the interest and experience of the individual doctor. I would interview any doctor you are thinking about using in regard to their experience with lupus and/or connective tissue diseases.
It is often a better choice to find a rheumy who practices at a large teaching hospital where they will see many different types of cases and share their experiences. That is not always possible and there are good rheumies in private practice but they can be hard to find in some areas.
Diagnosing most autoimmune diseases is not easy and it can take months or even years to get a confirming diagnosis and proper treatment. Labs often do not show up or can be variable. An experienced rheumy will look at clinical symptoms, labs, response (or lack of) to earlier treatments and make a diagnosis. It too me 14 months from the first sign of a problem to a firm diagnosis. However, my doc had already figured out what I had and was treating me aggressively before he got the final confirmation of MCTD in a lab test. Some AI diseases do not have a gold standard test and that is why training and experience is so important to diagnose and treat these mysterious diseases. Most doctors just do not have the requisite interest, training or experience to treat them. A good doctor should refer you to someone more qualified if he/she suspects a disease out of their field.
Your experiences are more the rule than the exception. I have talked a lot about both diagnosis and treatment hell.....most of us go through it to some extent and some never get out of it.