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Ken, you wrote:
"Two altitudes were needed, but there was no need for them to be of the
bodies used for the lunar distance."

Yes, exactly. From a "blackbox" perspective, this is true --almost any two
altitudes will do. At some time before or after, you take one sight, of any
object preferably near the prime vertical, to get local apparent time (the
famous "time sight") and you take another sight of any other object for
latitude (probably by a meridian altitude, but there were alternatives).
These are then carried forward (or regressed) to the time when the lunar
distance is measured. The time sight is carried forward by setting a decent
pocket watch to the local time and then adding or subtracting at the usual
rate of four seconds per minute of longitude for any distance you've
travelled in between (by DR). The latitude sight is carried forward simply
by adding or subtracting any change in latitude (again, as estimated by DR).
This was workaday stuff in the 19th century. Today we would recognize it as
the ordinary process of keeping a running fix. Then from the advanced local
time and the adjusted latitude, when you observe a lunar distance, you can
skip the process of measuring the altitudes of the bodies as long as you're
willing to put up with the extra work involved in calculating their current
altitudes. The altitudes taken with a lunar distance are really only
required to clear the distance of the effects of refraction and parallax.
Using them for anything else is optional. Calculating them is optional.
Incidentally, it's not hard to find 19th century navigation authorities who
will swear that it is "much more accurate" to calculate the altitudes. They
were mistaken. I suspect they confused the ability to carry the calculation
of the altitudes out to the last arc second with additional accuracy. Of
course since calculated altitudes were actually based on data derived from
earlier observed altitudes which were then carried forward, the calculated
altitudes would necessarily be less accurate, but generally not by much.

 -FER



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