I wonder what your professor's opinion on OED is. It includes shitake.

https://www.lexico.com/definition/shiitake (note the "also shitake")

The dictionary’s job is to describe you how words are used, not prescribe definitions.

Do you have any support for your position that that is a poor spelling? There are multiple ways in active use of transliterating Japanese. One of them involves writing shi+i as shii to produce shiitake, which is then left as is. Another one of them involves writing shi+i as shī to produce shītake. As ī involves a marker English does not have, it gets dropped when used in English. Both make sense.

Because double "i"s don't mean anything in English orthography, and at best serve as reminders of a sound you've heard somewhere else?

The name for that mushroom is not from a Roman script. It is translated into one in various ways, official and unofficial, devised by arbitrary missionaries. Once it becomes an English word, it's nice if it is respelled in a way that we can easily pronounce, although it's against our nature as Americans not to just leave it obscure and look down on people who aren't in the know.

British people will happily mangle a word to make it an English word, especially if it's French, but they really should be tried for what they do to the word "jaguar." Americans are more insecure, I guess, due to youth.

Isn’t that the same justification for Shitake vs Shiitake?

In Japanese it would be kanji or occasionally hiragana, some transliterations will be imperfect, and so long as the authors intent is clear, to me that’s all that matters