We might have to agree to disagree on that, because I'm not clear why you think the person claiming to be Ballard is no longer her because she was given some alien biologies and cultural influences. The ep makes it pretty clear that most of her original human personality survived and that she still wanted to be on Voyager, but that her alterations also made that difficult. I don't think it's comparable to Tuvix at all. By your definition, Sisko wasn't human because the Prophets caused his birth and interacted with him on an exclusive level so he could be their emissary.
Tuvix had the memories and personality traits of Neelix and Tuvok but he was a different being himself, more than the sum of the parts that went into creating him. The same was true of Ballard after her change; she had the same memories and some of the old genetics but had become something new. Over the time that the new beings that came from those seeds gained new experience and evolved in their new lives, they became a new life, new entities.
In either case, killing the new creature to recreate the old would be like trying to turn a butterfly back into a caterpillar. Except in both their cases, the caterpiller was dead before it became a caterpillar and new DNA was added to create the change.
'My' definiton doesn't make Sisko any different from any other human. the prophets maneuvered events to create a person who, to them, already existed and would exist. They caused 2 humans to conceive a child without adding any new DNA. In the case of Ballard and Tuvix, a new life was created from a pre-existing one with new genetics added to the mix. Tuvix wasn't even simply a Talaxian/Vulcan hybrid since the flower DNA was added in.