Two problems with this course of action: it sets a precedent ("OK that's Harry's. So where's William's?"), which means every Royal birth will have to be followed by a public release of maternity tests. Otherwise people will ask "why hasn't the test been released for him/her?" Almost as bad as having a government minister present to witness the live birth to ensure no substitution is made.
And secondly, no matter what the result of the paternity test, some would just claim it's been faked, there's a cover-up, a conspiracy etc etc, and nothing would be resolved.
The "Real or Fake?" issue would become a story in itself.
Best to leave it alone.