What I wrote could - should, indeed - be relatively clear to one who, like OP's DD, asked someone to read Judith Butler to back up her thoughts. I guess I thought this DD probably wouldn't understand it ...
But if you want to start to get to grips with the ideas around post-structuralism and hence Judith Butler and stuff, here's a start. (Butler is a really sloppy thinker, and often difficult to follow because of this ... and it's probably best not to take her or her writing too seriously; but, yes, post-structuralism can be tackled on its home turf, so to speak. This will be too brief. But, well, the history is interesting and the ideas surprisingly influential. So here goes.)
I mentioned the "transcendental signified". This is a coinage of Jacques Derrida. Two parts: 'transcendental', from Immanuel Kant and 'signified' from Ferdinand de Saussure (with an important hat-tip to Claude Lévi-Strauss).
Kant specified a form of argument he denoted 'transcendental' (he called his metaphysics 'transcendental idealism', although we won't need to do much with that). A transcendental argument uses the condition-of-the-possibility of something ... like this:
X is a condition of the possibility of Y
Y exists
(Therefore) X exists.
[Kant argued as follows, for instance: (the content of) synthetic a priori judgements (e.g. like those in mathematics) are a condition of the possibility of human experience; we humans certainly have experiences; so some synthetic a priori judgements are certainly well-founded. (That's what Kant called 'The Proper Problem of Pure Reason' solved, in a nutshell. Neat, no? Why it doesn't work? Left as exercise.)]
OK, now 'signified', an important term in semiotics (theories of signs and symbols) within the 'structural linguistics' (as it later became known) of Saussure - or simply 'structuralism', including the work of Lévi-Strauss on societal (rather than, as in Saussure, more narrowly linguistic) structures. (Possibly others too.)
Long story short: signs signify; two parts to this, the signifier and the signified. Enter Jacques Derrida. In order to have linguistic meanings, it seems we need to have signifieds for our signifiers. Otherwise it looks like our signs - including our words - will lack meaning.
The existence of signifieds, then, is a condition of the possibility of us meaning anything by our discourse. The signified is transcendental in that Kantian sense. So it seems, at any rate. But, as Derrida pointed out, Saussurean linguistics (and Lévi-Straussian social anthropology) locate signification only within structure - there seems no actually existing external-to-structure signifieds ('things', 'objects' as Ludwig Wittgenstein called them in his early work). Thus, Derrida concluded, there are no fixed meanings ... meaning is always ineluctably deferred (for which deferral Derrida minted a new word in French: "différance").
Hence Judith Butler and all that slipperiness about meanings and so on. (One aspect of postmodernism.) "What does 'woman' mean? - It's complicated ..." And off we jolly well go.
But there's another possible conclusion to be drawn from the lack of the so-called transcendental signified: perhaps signifieds are not transcendental after all. Perhaps we don't need things or objects for our signifiers to signify, at least in some cases, in order for our discourse to be meaningful. (I mentioned Wittgenstein earlier: he decided he was wrong about 'objects' wrt word meanings, and thoroughly and carefully debunked the whole idea in his later work. Of course lots of the postmodern crew, including Butler, misread Wittgenstein's later work, or simply ignore it, in this regard. Oh well.)
Of course this doesn't say there are no women, anything like that. No, really. The Supreme Court knew - as we all do - what 'woman' and 'sex' mean in everyday life and in law, without there being objects - signifieds - these words - signifiers - signify or refer to. Collapse of post-structuralist slippy-sloppiness and Judith Butler's nonsense.
[I haven't mentioned modus ponens/tollens. These are names of forms of logical argument. Look up "One person's modus ponens is another's modus tollens," to get the point I was making. I've written enough.]
I've over-simplified, but not much. Judith Butler and others do try to muddy the waters, it's true. But hold on to your common sense and you won't go far wrong. If OP's DD really does think Butler successfully justifies nonsense like "TWAW," "sex isn't binary," whatever, it really is just because she doesn't understand what's going on. (DD, I mean. -- Some people think Butler simply a charlatan peddling stuff she knows to be silly, but more charitably, I suspect she doesn't understand either. Who knows? It's obvious nonsense, though, when you read it, the more particularly when you see it in historical context.)