The sidereal day is a mean quantity, derived from many years of observations (it goes back to Halley times). But there are additional variations due to changes in the orbital parameters of the Earth. This changes are partly well known and predictable by celestial mechanics, but a significant part is unpredictable.
Google for a curve called Polar Motion. It depicts the movement of the north pole over the surface of the Earth through the years. Look for both the theoretical and the real one (build from measures). You will see the theoretical one is spiral-alike, but the measured "real" curve has a strongly erratic component superposed. It is even affected by earthquakes, and so is the exact position of the vernal equinox...
It has no sense to fine tune the determination of each sidereal day. It would be unrealistic, since the erratic, unpredictable contributions are in most cases bigger than the calculated corrections to the mean sidereal day.
Apart from that, I think there are some points in your reasoning that may be improved:
"Transit" is a well defined thing in astronomy. A line cannot transit. Moreover, the Sidereal Day is defined as the mean time between two transits of the Vernal Equinox. That point is, by definition, the zero of Right Ascension, always.
What you refer as "equatorial longitude" and "equatorial latitude" is very probably "Right Ascension" and "Declination", but I suggest that you carefully check that definitions, specially the sense (clockwise/counterclockwise) of your equatorial longitude vs Right Ascension, or you might probably run into a lot of problems when comparing your results with the literature.
In your calculations a factor cosĪµ appears and, although you may be doing something strictly correct from the geometrical point of view, I sincerely don't understand how you came to include the obliquity of the ecliptic in that calculations. You have probably projected some non constant movement from the ecliptic into the equator or something similar. Nobody can take you from doing that, but I think it leads unnecessarily to difficulties.
The ecliptic has nothing to do here. As long as we are not examining the movement of the real Sun, the ecliptic plays no role at all. It is the celestial equator the one that matters, since it is defined as the normal plane to the Earth rotation axis. There, in the celestial equator, all Hour Angles (and not "equatorial longitudes" or Right Ascensions, which are fixed) move with constant angular speed (as long as you neglect the 26000 yr periodic equinox precession and the regular + irregular movements of the north pole registered in the Polar Curve).
I kindly suggest you carefully learn the basic definitions and do lots of drawings. Thereafter, deriving relations will be safer.
A "year" without qualification may refer to a Julian year (of exactly $31\,557\,600~\rm s$), a mean Gregorian year (of exactly $31\,556\,952~\rm s$), an "ordinary" year (of exactly $31\,536\,000~\rm s$), or any number of other things (not all of which are quite so precisely defined).
Radioactive decay tables tend to be compiled from multiple different sources, most of which don't clarify which definition of "year" they used, so it is unclear what definition of year is used throughout. It's quite possible that many tables aren't even consistent with the definition of "year" used to calculate the decay times.
On the other hand, the standard error is usually overwhelmingly larger than the deviation created by using any common definition of year, so it doesn't really make a difference.
A day in physics without qualification pretty universally refers to a period of exactly $86\,400~\rm s$.
Best Answer
There seems to be some confusion. The number of solar days in a year differs from the number of sidereal days in year by 1--that difference of course being due the 1 revolution around the sun per year influencing the solar day.
Back to the number of days in a year: Baring tidal resonances, there is no reason for the length of a day to be commensurate with length of year; it is what is it: 365.2425
I remember this as follows:
365 day in the year
+1/4 A leap year every 4 years
-1/100 Except on years ending in "00"
+1/400 Unless the year is divisible by 400 (e.g. Y2K)
365.2425
so that 2000 was a leap-leap-leap year.