To determine the effect of changes in the local environmental conditions on Ptolemy's measurement of the Inclination of the Ecliptic I checked the record of conditions for Alexandria, Egypt at the times of the solstices in 2010 at wunderground.com. The correction factors for Bennet's refraction formula only gave a change of about 1 sec of arc less to the difference in refraction which is rather trivial if one is striving to achieve minute of arc accuracy for the Inclination. Only temperature and pressure corrections were included.
Bessel's Refraction Tables assume more complicated factors for the corrections for temperature and pressure. For more accurate measurements this might make a difference as well as the relative humidity. Since the Earth is not spherical but closer to a rotating ellipsiod changes in latitude might require a correction to the amount of refraction. Nitrogen and oxygen make up 99% of the Earth's atmosphere but that is for a dry atmosphere. Water vapor makes a 1% - 4% contribution near the Earth's surface where the atmosphere is densest and the most refraction occurs. This suggests a simplified model for the Earth's atmosphere.
Is the time of day of a solstice a critical factor? This year the Summer Solstice is at 1716 GMT and the Winter Solstice is at 0530 GMT. So depending on the time zone observations at the exact time of the solstice may not be possible. This year the Summer Solstice is at about 1:16 pm EDT which is a good time for Western Hemisphere including North America and Europe. Weather and timing will probably make the Winter Solstice more difficult to observe directly and one might have to do some sort of curve fit to get a good measurement.