How do we mathematically justify the straight line approximation of the calculation of Shapiro time delay?

While reading GR, I came about the tests of GR, one of them being Shapiro Time delay. The idea being that it takes more time to travel between the same two points of a null geodesic when a gravitational source is present, due to curvature effects. But while calculating the paths of the null geodesic in the presence of gravity, a lot of authors take the straight line approximation to the curved path. I want to mathematically show that such an approximation is valid. I know the arguments that the authors state, I just wanna do the calculation myself.

How do i show that the approximation is valid? And where can i find a derivation without the approximation?