My stutter is mild and I haven't really been in your situation before so this might be dumb advice but have you ever considered telling them ahead of time so they're prepared? Maybe tell them after the interview is scheduled so they are less inclined to back out? I kind of feel like if you tell them beforehand they'll be more mentally prepared for it. You shouldn't have to do this but if it gets you a warmer response maybe it's worth it? A lot of people just want to squirm when they are around someone that stutters. If they aren't used to it it can be difficult to handle. Heck, I stutter and even I have trouble with people that stutter.