i'm a medical student who stutters and i have been through this whole dreadful process so i can understand your concern. first of all, you're not alone. yes, i'm the only medical student i know who stutters, and everyday i deal with doctors and medical students who don't stutter. realistically, when you're in school, the feelings of isolation will continue.
that being said, i go through most of my day without any problem (by elusive techniques) but sometimes i just can't talk. i was so scared when i was applying to medical schools that i went onto google trying to find doctors who stutter to ask them of their experiences.
so first of all there are doctors out there who stutter. secondly, many of them stated that their patients found that they found them to be more humble and empathetic (a quality that is lacking in todays physicians). as you go further into medicine and when you read about all the diseases that people suffer from, from time to time you'll realize that what you have is minimal compared to many people. and then you'll stop reading, go back to work and after a while, you'll feel like crap again (it's a cycle)
that having been said, i think you need to learn to accept your problem. if you stutter, you will during your interview. it's a fact- you WILL be stressed and you will slip up. one thing you can do is come out and say that you have a problem early on in the interview because it will really take focus off of the problem and put it more onto WHAT you are saying, not how you are saying it. also it'll be clear that youre stumbling because you have a disorder, not because you're unsure or don't know what you're talking about. sometimes when i interview patients now, (depending on the situation) i'll start off by saying "before we begin, i just wanted to tell you that i have a speech impediment so in case anything i say is unclear, please just tell me because i want to make sure i get the appropriate information"
i think it's important to show that you are comfortable dealing with the fact that you have a speech impediment. that's why i'm not sure if it's the smartest thing to say that you find it to be a weakness. because realistically, it's a problem that you will have to deal with your entire life and you need to make sure that you're comfortable with it, because if you let it adversely affect your relationship with patients, it can have have bad consequences.
also, realize that there are many med students and doctors that even though they don't stutter, have horrible people skills, so the way you talk isn't the only factor- there are many many more factors that play into a patient's and person's impression of you. and you are going into a profession where your colleagues devote their lives to understanding people's disorders and problems- so give them the appropriate credit, they will understand and (hopefully) see beyond it. then there are people who may not see beyond it, but for the most part, i definitely feel like i'm being judged less among my medical colleagues. plus those people are usually dicks that no one really likes anyway.
as far as your interviews go, practice is always helpful, and as a person in your situation, i think the best thing to do would be to let go of your problem to an extent as you prepare and make sure you say exactly what you want to say when you go for your interview. i know that sometimes when i'm trying to be elusive and substitute words, etc, to hide my problem, i say something or imply that i didn't intend. this can really work against you in the interview.
then, practice saying what you want to say. also if you have already stated you have a speech impediment, i think it'll take a whole lot of pressure off of you. i did the hollin's program (HCRI) the summer after college, and i found it to be invaluable. they teach you specific speaking skills, which are very difficult to use all the time esp when you're so busy with school, but skills that are there when you really need them. again you probably won't be perfect, but at least a lot better.
anyway, my techniques worked for me. i got into a lot of great schools, had many interviews that went very well. when you are preparing for your interviews, let's face it, most of the people you are competing with don't stutter. you do, and you probably will. it sucks. the best thing you can do is focus on why you'd make a good doctor and treat your problem like any other problem someone might have. as far as getting through with the least amount of stuttering, you probably already know what works for you because you've been dealing with it all of your life (unless you're willing to invest in hollin's)
good luck, and if medicine is really where you want to be, then you'll get there.