Although it's possible that the dementors would not quickly report an intruder who gave the true invisibility cloak to a prisoner, if they have that degree of reasoning ability, I suspect they would take more notice of a person leaving one of the cells who they hadn't seen coming in than someone entering.

Why on earth would you expect them to wear it on the way in?

I don't recall Snape mentioning it at all though. Are you sure you aren't thinking of Quirrel?

It's in the chapter with Lesath. And it's not just that they know, but that they so casually mention it to Harry.

Let me rephrase that: In a work of fiction where jet-pack use was dramatically appropriate I would be disappointed with the smarts of 20 FBI men who never considered jet-packs when they were the only to their knowledge even slightly possible solution to a mystery, and in fact the correct solution.

I don't think we'll get much further in this particular sub-thread. It doesn't seem all that much anyway. Even if the cloak does not add all that much evidence to dementors = death it should at least cause Quirrel to generate that hypothesis, and we already know that even the commonly available knowledge was enough for Harry to correctly reach that conclusion. Considering that Quirrel has even more evidence beyond that he should as well, unless he has a mental blind spot.