You sound strong Frank. Good attitude. The best part is that you will do no worse in court than if you accepted her "offer". Of course the lawyers will be dining on steak and shrimp paid by you and her.
Keep an open mind. Remember, even here in bumbf*uk Alabama I got a judge who was as fair as he could be to men here. As others have said, you will be surprised.
As to the doctor thing, don't use that to accept less. Medicine is a big field and some doctors are not on call at all. Remember the job you almost took in the big city? Get the custody or the best schedule you can - then work it.
Chin up Frank. Your kids need you and deserve nothing less.