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I decided to take the opportunity to also look at Arcturus and below is a table with the facts.
 
The RA positions are fetched from Cartes du Ciel (although I am unfamiliar with the program and have not invested more of my precious time than absolutely necessary to obtain them):
 
  Sirius Arcturus Vega
2000 A.D. 6h45m11.31s 102.7 114.0 14h15m42.24s 216.7 66.1 18h36m58.11s 282.8 185
1000 A.D. 6h00m35.98s 91.5 113.6 13h28m43.64s 205.1 69.7 18h03m31.47s 274.8 182
0 5h16m23.27s 80.3 112.8 12h41m06.25s 193.1 73.3 17h30m26.88s 266.4 179
1000 B.C. 4h32m39.23s 69.2 111.2 11h51m35.30s 180.4 77.8 16h57m54.22s 258.2 176
2000 B.C. 3h49m32.26s 58.2 108.9 10h58m41.02s 167.1 83.0 16h25m55.84s 250.1 173
3000 B.C. 3h06m57.57s 47.4 104.9 10h00m28.75s 152.3 89.8 15h54m36.51s 242.1 171
4000 B.C. 2h24m35.78s 36.7 99.0 8h54m59.40s 135.7 98.6 15h23m53.93s 234.3 168
5000 B.C. 1h41m48.88s 25.8 91.1 7h41m06.65s 117.0 109.7 14h53m38.94s 226.7 164
98.6   76.9     99.7     56.1  

Going back in time must mean precession will diminish the RA values. Counting 7000 / 71 = 98.6 should give the difference between the RA days in 2000 A.D. compared to the RA days in 5000 B.C. However, the proper motions of the stars show that only Arcturus is relatively stable.

Sirius should move hand in hand with the Sun and 102.7 - 98.6 = 4.1 could then be the expected value in 5000 B.C.

Though counting with the Sothic year (equal in length to the Julian year) - 'twelve minutes longer than the duration of the solar year' - means Sirius will slowly drift behind the Sun. 7000 * 12 minutes = 84000 minutes = 1400 hours = 58⅓ days.

 
Comparing with my tabulated data in parallel with the glyphs, stretching down to the time of Bharani, it is obvious they cannot be factual but only the results of simplified extrapolations:
 
Ga3-21 Ga3-22 Ga3-23 Ga3-24 (84)
A Hydrae (144.1)

Vega

 Ukdah (145.4), κ Hydrae (145.5), Subra (145.8) ψ Leonis (146.4), Ras Elaset Australis (146.6) Vathorz Prior (147.9)
August 12 13 14 15 (227)
ºAugust 8 (220) 9 10 (*142) 11
'July 16 17 (*118) 18 19
'He Anakena 16 17 18 19 (200)
"July 2 3 (184) 4 (*105) 5
Castra (327.2), Bunda (327.5)

Sirius

 Nashira (328.0), Azelfafage, κ Capricorni (328.7) Enif, Erakis (329.2), 46 Capricorni, Jih (329.3), ι Piscis Austrini (329.4), λ Capricorni (329.6), ν Cephei (329.7), Deneb Algiedi (329.8) θ Piscis Austrini (330.1)
February 11 12 (408) 13 14 (45)
ºFebruary 7 8 (*324) 9 (405) 10 (41)
'January 15 (*300) 16 17 18
'Tua Haro 15 16 17 18 (383)
"January 1 (366) 2 3 4 (*289)

Which however is a reasonable way to approach the problem, because the Easter Islanders at that time probably did not have such great tools at hand as our modern computer programs (nor presumably records going back so far in time).