Actually I can relate to a lot of this except that I would be in your wife's role. Where I live there are ordinances regarding this dog situation. I'd probably call the police & they'd take care of it. Perhaps where you live that's not the case. Or perhaps you don't want to "rock the boat", as the saying goes. But I personally don't think anyone should have to be awakened at 5:30 a.m. every day by barking dogs. (We also have trains that run near our home. But I actually like to hear the sound of their horns at night.)

You wrote your wife has been diagnosed as having bipolar disorder and PTSD but she doesn't take med's & has stopped her therapy. Clearly you can't make your wife do what she needs to do to cope with her mental health issues. As DocJohn has written: "Denial is a Powerful Impediment to Treatment":

Denial is a Powerful Impediment to Treatment

My suggestion would be to see if you can strike a bargain. Tell your wife, point-blank, that you can't continue living with her mood swings (assuming you can't.) But you're also not interested in selling the house. (Selling the house you own & moving somewhere else is simply going to substitute a new set of anger-provoking problems for the one's you already have. The reality is this isn't about the house, the neighborhood, the dogs, or the trains. It's about your wife's inability to cope with the various kinds of day-to-day annoyances everyone has to deal with. I know because I have the same problem.)

Offer to go ahead & speak to the neighbor about the dogs (I know you don't feel you should have to... & you shouldn't.) But just do it in exchange for your wife's agreement to return to therapy & consideration of medications if it becomes apparent med's are going to be necessary to control her mood swings. (Perhaps the windows can be thrown into the bargain in some way or other as well. They sound like a good idea.) But if something such as that gets shot down out-of-hand, as it sounds like other things have, then perhaps what is going to be needed is for you to seek counseling or therapy services for yourself in an effort to figure out what you want to do. The cold hard reality here may just be there are not going to be any really good options, just a variety of not-so-good or even bad ones from which you are going to have to choose... if not now then at some later date. My best wishes to you.