To the extent that these might affect the apparent motion of the Sun as viewed from the Earth at all, the effect is negligible compared to the apparent motion of the Sun due to the Earth's rotation. For the purposes of our thought experiment, we can consider the Sun to be stationary WRT an inertial frame.
IOW, one may postulate the Sun as a hypothetically stationary object for his thought experiment and the math will work. He may transform coordinates.
But CMB measurements from the 1960's forward confirm that the Sun is not stationary. Nothing in the universe is stationary. Space/time itself is expanding.