But then the problem is not having the shared experience. If they did, they would just need to apply an arbitrary label,

Yes.. If they had the shared experience, they would just need to apply an arbitrary label, however given how we learn language(by association based on how they are used by people around us on what we see as objective events/experiences), I am not too confident the labels will match even after having the shared experience. My previous comment assumes this, but did not make it explicit. And I derive the

The reason why the thing can't be expressed is that it's too definite for language

quote from that assumption. I may be wrong about the assumption (since it seems to be more of a thought experiment than a practical experiment at the moment) but nevertheless I assign fairly high probability/confidence on that.