I mean, you joke, but existential dread might be an adaptive response to a hostile environment.

...a situation we might want to simulate for training purposes

I've written this in another comment but I'll repeat here. What you're asking really boils down to a combination of whether a human brain can be simulated and whether human intelligence is merely due to the physical brain (or do you believe in the existence of some intangible consciousness that cannot be replicated by a machine). Assume you believe both to be true, then your simulated brain is surely able to drown in existential dread because it's capable of no more and no less than the human one.

So you're defining "AI" to be anything that we can currently program a computer to do, and "I" to be anything we can't yet? That doesn't seem like a useful distinction to me. Unless you're using "I" to mean general (artificial) intelligence, in which case you should probably use the more well-known term.