If the "very accurate scale" is a balance scale then it will read the same anywhere on earth, on the moon, or on Jupiter. Changes in gravity effect both sides of the balance. A spring scale changes measurement when gravitational force changes.
All objects with mass have a gravitational effect. They do not need to be in our solar system. It gets smaller with distance.
Distance from earth to moon: 384,399 km (average)
Mass of moon: 7.342×1022 kg
G = 6.67408(31)×10−11 m3⋅kg−1⋅s−2
F = G m1m2r-2
Force is equal to the gravitational constant multiplied by mass #1 mass #2 and divided by radius squared.
F = 6.674×10−11 m3⋅kg−1⋅s−2 x 7.342×1022 kg x 1 kg / (384,399,000m)2
F = 3.316 x 10-5 kg⋅m⋅s-2
So near the equator the scale will increase and decrease around +/- 3.3 x 10-5 newtons with the lunar cycle.
Sun
1.496x 1011 m (average)
1.989x 1030 kg
F = 6.674×10−11 m3⋅kg−1⋅s−2 x 1.989×1030 kg x 1 kg / (1.496x 1011m)2
F = 0.00593 kg⋅m⋅s-2
F is in units of Newtons. So the "weight" of one kilogram is 9.8 newtons, 9.8 kg⋅m⋅s-2.
Rotation of earth has a centripetal force so a kg exerts less force on the spring. Near the poles the gravity from the moon and sun are close to perpendicular.