As you know, a fake nose is not a nose.
Why I think "that nose is fake" is nonsense is this:
"That nose" supposes there's a nose, but "that nose which is fake" supposes there's a fake nose, so "that nose is fake" suggests that there's something which is a nose and a fake nose at the same time, which is definitely contradictory.
What do you think? Do you think "that nose is fake" is nonsense? I think the correct version of it is just "that is a fake nose".
You understand what the sentence means; therefore, it is not nonsense. We can perform all sorts of analyses to try to analyse why the sentence is meaningful (e.g. the "nose" refers both to central members of the nose category, and to all vaguely-nasal objects, depending on context), but at the end of the day, if there is a sense in which a sentence is meaningful (and refers to real things, if you're using Wittgenstein's definition of "nonsense"), it is not nonsense.
Natural language is not formal logic. It is an error to try to treat it as though it were.
"Fake" in "fake nose" typically means the nose is artificial (e.g. man-made and surgically inserted), not that it's not a nose.
One might go one step further and say that it typically (but not necessarily always) means certain parts of the nose were artificially altered, added or removed, and that the parts of the nose that relates to the nose-being-ness of the nose weren't changed. In that sense, "fake" would just be describing some parts of the nose.
Also, one could use "that nose is fake" to mean "that object there that looks like a nose is in fact not a nose", which would not be contradictory.
By your argument a "rubber duck" is nonsense too, as is a "prosthetic leg".
Avoiding this type of pedantry/sophistry just renders human conversation too tiring to bother. Anything that reeks of figurative speech (metaphor or metonymie or even exagerration), becomes outlawed. Any word has to be justified with exactly which author's definition of each term/concept you're using, because they will always contradict at some level --- I read a PhD thesis listing more than a hundred (all at least slightly different) biological species concepts that fell into roughly ten groups; so you can't use words like "type" or "species" anymore. If that is the way you lean, Lojban is probably the only way to go (however).
Language probably evolved on the basis that we name similar things with the same signifier, that's how kids pick it up; anything that connects more than two wheeled units will be called "train" whether there's tracks or not -- and often already if it has two.
In human languages, anything duck-like will be called a duck, whether a waterbird or a wooden decoration or a plastic (not rubber!) bath toy.
So that prosthetic thing on your face that looks like a nose... it's a nose.
Wittgenstein spent a lot of time on this sort of thing. Is the word "cat" a cat. No. Is a photograph of a cat a cat? No. Is a fake nose a nose? No. But the statement "That nose is fake" is not nonsense in natural language, because it has a meaning.
As you know, a fake nose is not a nose.
I disagree. You seem to conflate the definitions of "nose" and "genuine nose"; which lies at the basis for your claim, but I'd like to counter that.
Consider this generalized statement:
[adjective] [nouns] are a subset of [nouns], specifically the [nouns] to which "this [noun] is [adjective]" is considered to be a true statement.
Simply put, green balls are balls, furry animals are animals, smart people are people, and fake noses are noses.
I am aware that you can find apparent contradictions such as a "red herring" not necessarily being a herring (i.e. the fish), but that is beside the point. The argument being made by the OP is one of logical definition, not one of occasional homonyms, idioms or figures of speech.
"Fake nose" does not mean "not a nose", it means "not a genuine nose". I suspect you're getting confused because we generally allow the "obvious" adjective to be omitted when it is either commonly inferred or irrelevant for the topic at hand.
Secondly, I want to address the level of scrutiny that you're trying to apply to the language, while at the same time playing fast and loose with questions like:
What do you think?
To which the pedantic answer is "thoughts".
However, I suspect you were relying on the obviously inferred interpretation of you asking us what we think about your interpretation of this phrase, except that you omitted this last part from the question you actually wrote.
Clearly, you allow for shorthand speech when the intended meaning is obviously understood by the listener, since you applied this approach in your own question.
Even though I've already disagreed with your claim that a "fake nose" is not a nose, let's entertain your claim, while also factoring in the permissible shorthand speech that you've used. In this scenario, there are still possible interpretations that are not incorrect in the way you claim that it should be. The first few that come to mind are:
With the right surrounding context, "that nose is fake" can be applied to all of these scenarios without being incorrect.
Would you consider the part of a statue that is in the middle of the face and is meant to look like a nose and contains nostrils to be a 'real' nose, or even a nose at all? What about Frosty's 'button nose'. If you want to be precise, you have to define your terms precisely.
If someone was in an accident which destroyed their natural nose and is using a prosthetic for a nose, what would you say? It is a prosthetic nose, it isn't a real nose. But it is a real prosthetic nose. So it it real? What do you mean by real? Words have different definitions depending on context.
When you say it 'isn't a real nose' I would take that to mean that it isn't the natural nose that grew with the face, using this definition:
(of a substance or thing) not imitation or artificial; genuine.
For 'fake' I would use this definition:
not genuine; counterfeit
You are assuming a definition of 'real' that you are not specifying:
actually existing as a thing or occurring in fact; not imagined or supposed.
