Hi Jup,
Sirius is a strange star, hardly effected by procession in the Julian Calendar, as shown below:-
"It has been noticed, and the Sothic cycle confirms, that Sirius does not move retrograde across the sky like other stars, a phenomenon widely known as the
precession of the equinox. Professor
Jed Buchwald wrote "Sirius remains about the same distance from the equinoxes—and so from the solstices—throughout these many centuries, despite precession."
[11] For the same reason, the heliacal rising or zenith of Sirius does not slip through the calendar at the precession rate of about one day per 71.6 years as other stars do but much slower.
[12] This remarkable stability within the solar year may be one reason that the Egyptians used it as a basis for their calendar. The coincidence of a heliacal rising of Sirius and the New Year reported by Censorinus occurred about the 20th of July, that is a month later after the summer solstice."
The Julian Calendar was constructed on 1st January 0045 BC, but the Greek Pharaoh Ptolomy III tried to add a leap year to the Egyptian Calendar, but failed because of religious reasons in 0238 BC.