Wendell Brunner, you wrote:
"I had never looked at Frank Reed's methods when I decided to try a lunar in 1992 on my voyage to Hawaii"
Yeah. Well. I had no methods for lunars in 1992, but there were other resources, including old editions of Bowditch, Norie, etc., and, for a 20th-century resource, Letcher's "Self-Contained Celestial Navigation" which had excellent material on the subject. Of course digging into an obscure corner of navigation history, like lunars, was much more difficult in the pre-Internet days. But that's all in the past...
"I tried to figure out myself how to do it. If you had the lunar distance, sun and moon altitudes all measured simultaneously, it would be pretty straightforward. I remember reading in one of the Hornblower books that it took three midshipmen observing together to do a lunar, but that was clearly not how Slocum did it."
Horblower was fiction. As for Slocum, he was a footnote in the history of lunars. He wrote a fine epitaph for lunars in his "Sailing Alone Around the World", but his experience wouldn't have helped you much. In any case, you can find out how a single observer would do the work in most old manuals of navigation. You shoot the altitudes before the lunar distance sights, then shoot them again after. It takes only simple averaging to synchronize them with the distance observations.
Many modern navigators, accustomed to the frantic timing of observations of celestial altitudes, make the incorrect assumption that the altitudes must be measured with great accuracy and careful timing, but that's all wrong. There's nothing difficult here. In a typical sequence, you would shoot the Moon's altitude, then the Sun's, followed by three to five lunars. Then shoot the altitudes in reverse order, Sun first, followed by the Moon. Average the lunar distances. They fall to the central time among them. Average the two Sun altitudes. They, too, fall to the common mid-point time among the sights. And last average the Moon altitudes. As with the others, they fall to the central, common time. What you're left with is the basic set of observations required of any lunar: Moon altitude, Sun altitude, and Lunar Distance, all at a common time.
You also wrote:
"When I observed for my lunar, I measured a series of Lunar Distances, sun altitudes and the time between those measurements, but I couldn't get the moon altitude because the horizon under it was obscured."
Well, that's unfortunate, I guess... But lunars are not demanding. You don't need a set "right this second". So if there's no horizon, you wait a while until conditions are better.
"I took the data home and despaired of a solution, until I realized that longitude, time and lunar distance were linked, and a lunar distance along with any other celestial observation was enough to determine the solution."
Up to a point, yes, this is true. You can always feed a "DR" position into the process and use calculated altitudes. Then if the longitude doesn't match up at the end, you repeat the process. Calculating altitudes like this was sometimes recommended in historical sources, but it was a lot of extra work for dubious benefit. It created an illusion of accuracy for some computation-focused navigators. Those navigators were confused. In the modern world, if you want to try lunars, it all depends on your goal. You have to settle that first. WHY are you shooting lunars, at all?? For some circumstances, trial and error (iteration) on a type of "assumed position" may be valuable. But again, you have to decide at square one, WHY are you shooting lunars?
"The iterative method allowed me to calculate the solution, and then I realized it had some other advantages as well. The iterative method allows one to focus on careful observations and let the calculations handle the other complications."
The reason we shoot the altitudes in lunars is because it is easy to do so. It's EASY to shoot the altitudes. It spares us the effort of doing the calculations. And it spares us from having to make any major assumption about position. Point being: observing the altitudes is not a bad thing, not an addition to the effort. It's a good thing, done for practical benefit, that reduces the workload.
"You don't need three midshipmen."
That's so true. And you never did. Some very early navigation manuals (and fictional accounts based on them) recommended having a pair of assistants at the ready to shoot the altitudes when the distance was taken by an experienced observer. This was advice for observers on vessels with abundant spare labor. But it wasn't necessary. It was repeated in fiction probably because it sounded ceremonious and impractical. That's not a reflection of reality.
"I think it would be pretty tedious to do this calculation with logarithms, but with a cheap calculator it is reasonable."
There are many ways to do the calculations using a calculator. It could also be done with paper methods. Your trial-and-error (iterative) approach is only one narrowly-specified way to proceed. Why use it?
"The main advantage of this method is that you don't have to try to determine a lunar distance and sun and moon altitudes simultaneously, something that I think is observationally awkward."
No, it's not "awkward" or difficult to do at all. This was normal practice for decades. Of course for a modern observer, circumstances are quite different. What are your goals? If you're just experimenting with lunars to experience some aspects of the historical observation, then shooting the altitudes can seem like wasted time. You already know how to shoot altitudes, so where's the fun in that? Also if you're shooting lunars inland, the altitudes are much more difficult to get (you need an artificial horizon and both altitudes below 60°). Of course modern observers experimenting like this generally know their locations exactly, so just enter that position and let the app calculate the altitudes. That's exactly what happens in my online lunars app. You can also iterate to a position using my online app. You have to decide, what is your goal? Why are you shooting lunars?
It's very cool that you worked out a unique methodology for yourself, by yourself, back in the early 1990s, but we can do better today. We have been capable of better for nearly twenty years. We can do better with modern goals for lunars. And we can do better with historical goals for lunars.