Island sink drains... slowly
As suggested, I'm posting the (accidental) resolution as an answer.

Update: The problem appears to have resolved itself, and it appears to have been some sort of very minor obstruction - not in the drain, but in the garbage disposal. At the time I posted the question, I had finished the plumbing but not the electrical wiring for the island; the first time I ran the garbage disposal, the drain ran freely and hasn't had an issue since. I still don't understand the physics - why would removing the AAV have had an effect when the garbage disposal was the issue? - but I'm satisfied that the problem isn't/wasn't a venting issue.