I don't see how it causes the argument to fall apart.
Apparently so. Alright, then. Is a transplanted kidney then part of its new host? Yes, by pretty much every measure and effectively no one sane would consider it to be a separate person itself. Is the DNA the same as the host? Mostly, even if it's from a completely unrelated person, but not entirely. This alone is plenty to demonstrate that "person" is a distinctly different concept from DNA. That alone picks at a very serious flaw in how you're trying to equate person and DNA.
You just raised crime scene DNA testing. That's plenty useful for what it does, but it's also quite irrelevant for the concept that you're trying to defend. If the DNA from that transplanted kidney ends up being tested, does that somehow change anything about which person did what? No, of course not, even though the DNA is not the same as the person's original DNA. At best, it might confuse the issue a little and potentially lead to false conclusions - which is little more than what you're working to do here. For the DNA purposes here, a zygote works in a way that's little different than a transplanted kidney (though it's rather likely to have DNA more similar than the kidney to the host, even if that not necessarily the case - a transplanted identical twin kidney is still a kidney from another person, even if it has effectively the same DNA), albeit not really beneficial to the host, unlike a successfully transplanted kidney. The main difference there is in its potential future development, rather than its then current state.
There are deeper questions here, though, such as what qualifies something to actually be a person, much less a living person? DNA is useful for distinguishing purposes when there's already a number of other important premises met that qualify those involved as persons, but is not sufficient in and of itself to determine personhood. Frankly, it probably also wouldn't be wrong here to take things back to the fundamentals of biology and what a human being actually is in the end, but again, that's so much more messy than we tend to like to think.