I wonder if it'd be more likely for an uncommon animal product to have kanji or kana. Like would the packaging for alligator/crocodile be ワニ or 鰐.
I think because so many fish are commonly eaten people get used to a lot of the kanji but anecdotally there are lots that are still very common in kana on food packages like イカ、ヒラメ、ソイ etc