Greetings and thank you for your question. I'm sorry your son and you, his parents, are going through all of this! Bipolar is a tough (but not impossible!) disease to treat. Part of the challenge is that many people with bipolar refuse to acknowledge that they have it, or refuse to get treated for it. I'm not sure which category (or both!) he falls into, but the fact of the matter is, until he becomes willing to get and stay in treatment, the options to you, his parents, are very limited.
Certainly, you can apply pressure financially (i.e., "Get treatment or you'll have to move out") but those strategies often fail or backfire. Tough love is hard to apply when the person already feels miserable.
One way to help him is to encourage him to get back in with a psychiatrist, but a different one. You can help him in this regard by asking around for recommendations from people you know: do they know any psychiatrists who treat patients with bipolar disorder and are the kind of personality that would fit your son? Get a list of names, contact their offices, and see gow they "feel" to you over the phone. Of course, it is up to him to actually go, but if you have the legwork done and also can say that someone you know goes or trusts the person, that may encourage him to go.
Also, have him do his own research into bipolar, blogs from other patients with it, etc. The more he sees that other people have had it, gotten treatment for it, and done well, the more it will encourage him to seek treatment, too.
"An Unquiet Mind," by Kay Jamison, is a must-read for you and him. She is a professor at Hopkins who suffers from bipolar. The book is an autobiography of her struggles (including her many moves overseas!).
Next, reassuring him that this is a disease, just like diabetes or asthma, is important. It was not his choice to have it, but it is his choice what to do with it. And like those diseases, it is quite treatable, with treatments that don't have to be horrible or limiting.
Ultimately, he must decide. But these are places you can begin or focus upon. Best wishes to you all! Please let me know if I may be of further service.