Sun's gravitational effect: Lightarrow, in addition to the acceleration to the Sun's gravity - which gives you factor of +/- 40g effective weight, there is the acceleration due to the circular motion of the Earth's orbit around the Sun. I remember I did some sums a long while ago. The effect of, shall we call it centrifugal force, is in the opposite direction to that of the gravity and more or less cancels out. In the same way that an astronaut feels weightless when in orbit around the Earth, the kg mass and the Earth are both in 'free fall' around the Sun -so there is much less difference than the 40g. I calculated the difference to be about 1/1000 of that, taking both accelerations into account.

I was, originally, trying to work out the effect on tides but gave up when I realised that the tides actually consist of a wave which sloshes around the Earth once a day (ish) and the actual Q factor of what is, in effect, a resonator, would be a major factor in determining the actual heights of tides.

I calculated the sums of the rotational and gravitational effects due to the Solar orbit and the Lunar 'wobble'. This suggested that the difference due to the Moon's effect would be about 4e-5 N/kg whilst the effect of the Sun would be about 2.6e-6N/kg. It would mean a variation in measured weight of 4e-4% for the moon and 2.6e-5% for the Sun.

This isn't far from the ratio which they quote for the relative effect of Sun and Moon on the tides.