"mistrust with" is indeed incorrect.
"mistrust of" is correct but it is not an idiom, and you shouldn't blame your getting this question wrong on "idiom errors." an idiom is usually a grammatically incorrect phrase that is so commonly used that it is accepted.
you typically have to memorize idioms to know that they are correct.
knowing "mistrust of" requires reasoning, not memorization.
mistrust is an action. you put "of" after an action in order to indicate something that is acted upon.
the killing of Jane ---> Jane was killed
the mistrust of her findings --> the findings were mistrusted, or not trusted
in other words, knowing these phrases is not as simple, or rather, as difficult, as memorizing them. "of" is a preposition that has many definitions. you simply need to apply the correct definition by reasoning, not memorization