I could be wrong, since I'm not a doctor and all, but it's my understanding based on about ten years of studying issues like bipolar disorder online that to be diagnosed with bipolar disorder, as well as other disorders, like OCD, people merely need to have a set of symptoms, which disrupt their lives sufficiently, which hopefully respond to certain kinds of treatment (and of course, they probably need to be diagnosed in order for them to officially have it). This, as opposed to a specific physiological and/or genetic condition that causes or might cause the symptoms. It's not like strep throat where you can test for bacteria for a definitive diagnosis. There could be infinite causes of bipolar disorder as I understand it.
Thus, any tests that could diagnose it might miss a plethora of cases, and many of the people diagnosed by those tests might manage their condition perfectly already (which means they don't have bipolar disorder).
It's my understanding that the aim of such tests would be to assist doctors in making a good diagnosis rather than to replace the doctor's decision.
Nevertheless, there could be other conditions with symptoms similar to bipolar disorder, which do have a definitive underlying cause: e.g. vitamin B12 deficiency. I guess it's probably up to the doctor to decide if the individual also has bipolar disorder.
Anyway, I'm not saying bipolar disorder is all in one's head. At least some of the symptoms are physiological (such as how patients tend to get more manic if they take anti-depressants). I'm sure a lot of doctors use that criteria to help them diagnose the condition already. While I don't know another condition that would match that issue, it's possible there is one. However, just because symptoms are physiological doesn't mean they always have the same cause, which means they could have different methods of diagnosis.
Now, all that being said, while I still think a definitive test is possible, a new way of testing would perhaps be needed. A blood test isn't going to tell you behaviors or how well a person is managing their symptoms, or if the person is using good coping mechanisms that might cause them to not need a diagnosis. A blood test may tell you things that predispose people to behaviors and symptoms, however.
I wrote this answer before I read Bryan Crause's (I just read it). So, sorry that I repeated some of it.