A small, polar-orbiting moon

I'd like input from all you orbital mechanics out there as to this
concept's feasibility:

What if sometime in prehistory, the Earth had captured a Near Earth
Orbit asteroid, say 10 km in diameter, into a nice circular 20310.8
kilometer polar orbit?

So every 8 hours the Moon2, let's call it 'Cynthia' [derives from
'woman from Kynthos', a reference to Artemis, sister of Apollo and
Greek goddess of the Moon, who was reputed to have been born on the
mountain of Kynthos]
rises from either the Northerly horizon or the Southerly and, in
about 3 hours or less, sinks below the opposite horizon. The ancients
would quickly figure out the regular patterns of its orbit and how to
use it to determine longitude (using only the local time of day; no
accurate clocks synchronized to Greenwich time needed). Accurate maps
would appear early. The apparent size of Cynthia would be only about
1/12 that of Moon1 (Luna) and would vary by about 5%, depending on the
time and place of the observer. The size variation would let the
ancients determine the distance to Cynthia and start giving a sense of
the scale of the solar system. The ancients would probably deduce
that Cynthia was brighter (relative to size) than Luna because it's
closer to Earth. So they'd estimate the distance to Luna and, when
they compared the orbital periods of the two moons, would derive the
gravitational inverse-square law centuries earlier than OTL. Eclipses
(solar transits, really) would happen often, encouraging study of the
Sun's surface (maybe using camera obscura, really big pinhole camera
minus film). Cynthia often being 25 times closer to us than is Luna,
naked-eye observation would show craters and such on Cynthia, an early
intro to Galilean ideas of imperfect heavenly bodies or, better yet,
the idea of other planets and moons being actual places, that is,
destinations.

There is some question as to how bright Cynthia would be. I'd
appreciate any critiques of the following analysis:

… 1) For the non-astronomers out there, the brightness of
celestial bodies/stars/whatever is given as 'visual magnitude'. For
some reason, early astronomers defined that an object of magnitude 1
was 100 times brighter than an object of magnitude 6. So an object of
mag 1 would be 2.512 times brighter than an object of mag 2 because
2.512^(6-1)=100 and 2.512^(2-1)=2.512. Setting magnitudes was an
attempt to compare the brightness of stars but later really bright
objects were assigned magnitudes as well. For example, Luna has a
magnitude of -12.5 (brighter objects have larger negative numbers).

2) Let's wave our hand and make Luna (diameter 3476 km) disappear and
be replaced by Cynthia (diameter 10km). The area, and so the amount
of light reflected, has shrunk to (10/3476)^2 or 0.000008276 of its
former value. Its magnitude has changed by 12.7, meaning it's dimmer
by 2.512^12.7 times. Its magnitude is now 0.2 (-12.5+12.7), like a
really bright star but still not visible in full daylight.

3) Now let's move Cynthia closer to Earth. At closest, Luna is about
378028 km from an observer on Earth, while Cynthia would be about
13938.8 km away, 27.12 times closer. So it's brighter by a factor of
735.5 (27.12^2) due to the inverse square law. This produces a change
in magnitude of -7.2 (2.512^7.2~735.5) so Cynthia, at best, would have
a magnitude of -7.0 (-7.2+0.2).

4) By comparison, Venus can have a magnitude of -4.9 while Luna, as I
mentioned before, has a magnitude of -12.5. So Venus Cynthia
Luna.

5) From sci.astronomy.amateur, "The Great Comet of 1744 reached -7.0
magnitude and was visible 12 degrees from the Sun in broad daylight."

So I think Cynthia would often be visible during the day and available
for use in navigation.

My greatest concern is whether/how long Cynthia might maintain a
stable, circular polar orbit. Would the influence of Luna disrupt her
orbit? I assume the orbit would precess but how fast? Would the
precession be at a constant rate, one that ancients could include in
their tables/calculations? Is it even possible for a Near Earth
Asteroid to take up a circular orbit around Earth?

Thanks in advance for any input.