The first definition of ’かしら' on goo.ne is "人間や動物の首から上の部分。あたま。こうべ。「尾頭つき」人間や動物の首から上の部分。あたま。こうべ。「尾頭つき」" which roughly says "The part above the neck for a human or animal." So this is 'Tail'-'part above neck'-'including' which while yes, 頭付き seems to be used to mean 'fish with head included', in this case it's saying 'everything from tail to above the head included'.
In the context of the story, I can't see that being anything other than a play on words about the fact the MC comes with a tail. Unfortunately, you'll never get any of that without even mentioning 'tail' in the English translation!
So yeah, unless someone has a link to the author explaining why this wasn't a play on words and why she was, in fact, talking about how she likes to buy her fish, I think we should leave the alternate name as this new one.
This phrasing doesn't even apply only to fish; it's just the most common. Definition says animal, so it could mean something like, I don't know, maybe pig?