It's definitely a tough situation your in. Wanting to excel in your career and move up but also feeling like your speech is impeding your progress.. I think this is a common problem and concern for people who stutter. How we fit in, in the workplace and do we fit in at all. I think the only thing you can do is be open with your colleagues and your manager about what your dealing with. Turn your skeptics into your supporters and encouragers. If you open up to people and show that you too have something you struggle with on a daily basis then people tend to change their perspective and how they view you because their equipped with new information. I remember hearing about the same thing from another stutterer. He has a YouTube channel where he runs a podcast interviewing other stutterers. And in some of these videos he talks about his experience of doing regular 4 hour presentations for his company. Apparently he's a trainer of some kind. He presents to new people every month and he before he starts presenting he always says something to the effect of "Hey guys, I have stutter so you'll notice it may take me a couple years to say some words. Feel free to help a brother out if you know the word." Like something light hearted like that. I found an instance of him sharing his experience and personal take on the matter, he probably says it best. Skip to 23:12 https://youtu.be/HaI2julIbeA?feature=shared