I knew this pastor who was a native English speaker and a stutterer. In English he stuttered almost every word but, he was fluent in the local language and he could give the Sunday Sermon without even skipping a beat. If you are bilingual, after you reach a certain point of fluentness in the second language, your brain sort of stops translating and just comprehends each language on its own. Imitating a different accent might be enough melodic difference in the language to offset the stutter. (Like British if you're American).