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Consider two different approaches below.

1. Use special relativity and compute time contraction due to velocity. Use approximation of general relativity in the Newtown limits and compute time expansion due to the less gravity and then find the total time dilation.

2. Don't use special relativity. Stick to the approximation of general relativity based on the symmetry and find Schwarzschild metric and the geodesic for the Earth limits. Find the time dilation assuming a relative velocity in the metric.

The question is:

Which of them are more justified and provide a better approximation? Are they equivalent? What happens when the relative velocity of the satellite is zero?