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3. Be that as it may be. We should continue with Nunki and according to my astronomy book it was at that time rising heliacally just after 19h:

20th hour:

Nunki σ Sagittarii 2.05 21° 49′ S 19h 01m 289.4
  ι Lyrae 5.25 36° 06′ N 19h 05m 290.5
Aladfar η Lyrae 4.43 39° 09′ N 19h 12m 292.1
  θ Lyrae 4.35 38° 08′ N 19h 14m 292.8
Ga8-21 Ga8-22 Ga8-23 Ga8-24 Ga8-25 Ga8-26
λ Lyrae (287.7), Ascella (287.9), Nunki (288.4)   ι Lyrae (289.5) Aladfar (291.1) θ Lyrae (291.8)  
'May 26 (146) 19h (289.2) '28 '29 '30 '31

At the time of G Nunki was presumably rising a day earlier, in 'May 26. Right ascension time 19h corresponds to day number 19 / 24 * 365¼ = 289 beyond spring equinox north of the equator and 80 + 289 = 369 points at 'January 4.

The end of the year ought to have been in the night following the heliacal rising of Double Double (ε Lyrae):

Ga8-10 (*277) Ga8-11 Ga8-12 (216)
  Kaus Medius, κ Lyrae (277.5), Tung Hae (277.7), Kaus Australis (278.3) Al Athfar (278.6), Kaus Borealis (279.3)
'December 23 '24 '25
"October 21 "22 (295) "23
Ga8-13 (*280) Ga8-14 Ga8-15 Ga8-16
    Vega (281.8)  
'26 (360) '27 '28 '29
"24 "25 "26 "27 (300)
 
Ga8-17 Ga8-18 (222) Ga8-19 Ga8-20
Double Double (283.7), ζ Lyrae (283.8), φ Sagittarii (284.1) Sheliak, ν Lyrae (285.1) Ain al Rami (286.2), δ Lyrae (286.3) Alya (286.6), ξ Sagittarii (287.1), Sulaphat (287.4)
'December 30 (364) '31 'May 24 (144) '25
Ga8-21 Ga8-22 Ga8-23 Ga8-24 Ga8-25 Ga8-26
λ Lyrae (287.7), Ascella (287.9), Nunki (288.4) Manubrium (288.8), τ Sagittarii (289.4) ι Lyrae (289.5), Al Baldah (290.1) Aladfar (291.1) ψ Sagittarii (291.6), θ Lyrae (291.8)  
'May 26 (146) 19h (289.2) '28 '29 '30 '31

The Sagittarius star at Ga8-19 is Ain al Rami (ν) and we remember ain means eye. The 'eye' in Taurus (ε) - the Bull's Eye (cfr at Coma Berenices III) - comes 223 - 3 = 220 days earlier:

Gb8-30 (*64) Ga1-1 Ga1-2 Ga1-3 Ga1-4
Hyadum II (64.2)   Ain, θ¹ Tauri, θ² Tauri (65.7)   Aldebaran (68.2)
'May 24 '25 '26 (146) '27 '28

At Hercules I suggested a connection with Zhōngshān (ο Herculis) and indeed we can see the pupil of the eye:

"ν¹ and ν², red stars of the 5th magnitude, 12' apart, and both double, were Άin al Rāmi, the Archer's Eye.

Ptolemy catalogued them as a nebulous double star, - νεφελοειδής καί διπλους, - among the first to be so designated.

With ξ and ο [Manubrium] they were the Chinese Kien Sing, a Flag-staff." (Allen)

I imagine the star name 'Handle' (manubrium) could refer to the central bone in front with this name:

... The cope (known in Latin as pluviale 'rain coat' or cappa 'cape') is a liturgical vestment, more precisely a long mantle or cloak, open in front and fastened at the breast with a band or clasp ...