“Sez who?” sez who.
My bad – I did miss that.
I’ll leave the punctuation judgment call to others.
It’s the punctuation that makes it a sentence fragment, isn’t it? According to the OP, the two fragments have to be identical.
While I’m at it with the boring, routine stuff:
“when appended to its own quotation forms a sentence which Victim’s Name Here doesn’t think is true” when appended to its own quotation forms a sentence which Victim’s Name Here doesn’t think is true
I can correctly think or not think this sentence is true, and so can you, and everyone else in the world, except for poor Victim’s Name Here; whatever they do, they’ll get it wrong. [But for the rest of us, it’s easy; just take the opposite position from VNH, of course]
‘Is a sentence fragment’ is a sentence fragment.
‘Isn’t a complete thought’ isn’t a complete thought.
What do you mean by ‘what do you mean by’?
How do you say ‘how do you say’?
‘Would make a crappy title for a story’ would make a crappy title for a story.
‘Could be clearer’ could be clearer.
“Is a valid quine” is a valid quine.
“Is not a valid quine” is not a valid quine. (I’ll leave the meta-logic of this one alone 
)
Thread over - we have reverse quines!
“Didn’t know such a thing was possible” didn’t know such a thing was possible.
“Has twenty-nine words (counting hyphenates as two), five commas, and the numbers twenty-nine, two, and five, and isn’t a stand-alone sentence because it lacks a subject” has twenty-nine words (counting hyphenates as two), five commas, and the numbers twenty-nine, two, and five, and isn’t a stand-alone sentence because it lacks a subject.