I feel like since you're so new, if you mention anything about needing a shift accommodation it might not go over too well or be in your best interest just because you're still in a probation period which I think means they can let you go for basically anything. I'm sorry you're going through this depending on how important this job is to you, I wouldn't risk it. I'm not sure what I would do in this situation if I were you but I think asking for a doctor's note would be the last thing I would try. Because if you don't want them to know about any type of health issue, that would be a pretty obvious sign. Maybe just wait and see if you get called in for a third shift, then ask other coworkers to take it, or talk to your boss about it personally, but again I'm not sure if that would go over well. Maybe you could just do the 3rd shift occasionally and make it clear that you really want 1st or 2nd shifts.
I don't have much work experience so I have no idea how to handle this situation, but since you said you don't want your boss or coworkers to know about your BP, giving them a doctor's note would be basically the same thing as telling them. Plus they might just find someone else since you're so new.
I think I would wait awhile and take it a day at a time to feel out how well I manage a 3rd shift and if I absolutely couldn't after one, then I would get the note, just be prepared for it to be brought up and discussed and have the potential of being let go. I'm sorry