I suppose "satisfactory" is the key word in what you wrote. Certainly there have been possible explanations offered, particularly with regard to REM sleep (Exploring the Connection Between Sleep and Cluster Headache: A Narrative Review - PMC).
According to one study (attached here), 2am is the most common time for attacks, but midnight, 1, and 3 are right up there. (Rather than a specific time, it's probably more accurate to say ~90 minutes after falling asleep.) (At least as far as I can tell, the attached study actually tells us almost nothing, since of course people get multiple attacks and they all are counted here.) However, as many as 25 percent of people with CH don't get nighttime attacks, and the number of reported attacks (in the study) at 2pm and 3pm (outside the time range of your theory) isn't all that much lower than the number at midnight or 1am.
Well, but why would it want people to be awake and experiencing excruciating pain, or use excruciating pain to wake them? And why, if maybe we all, or most of us, have a biphasic instinct wired into our brains, are so few people afflicted with CH (or just generally, why do so few people wake up, with or without pain, at those early-morning hours, or ~90 minutes after falling asleep)? And why does a daytime nap so often bring on an attack? These questions are not to dismiss your idea, which I think could be part of the puzzle, but to say that there are surely more complex things going on and to agree with you that it would be nice if there were "a neurologist specialising in both sleep and cluster headaches out there who latches on to this."
Of course, there are whole cultures that have a biphasic sleep pattern, though not exactly the kind you have been referring to. A siesta in the afternoon along with some nighttime sleep (usually less than 8 hours) is pretty common in many Latin American and European countries. I guess someone could look at CH incidence among those populations.
Rozen - Cluster_Headache_in_USA-2.pdf