There are probably a lot of ways to do this.
Note that the parametric equation of the sphere is very similar to the parametric equation of an spheroid. If you can calculate the geodesic of a sphere from its parametric representation, you might be able to apply the same procedure to the spheroid.
http://www.cs.iastate.edu/~cs577/handouts/geodesics.pdf
You could also use the tools of tensor calculus by calculating the Transformation Matrix from Spherical To Oblate Coordinates $[\frac{dx^a}{d\bar{x}^u}]$, and then calculate the oblate metric using $\bar{g}_{uv}=\frac{dx^a}{d\bar{x^u}} \frac{dx^b}{d\bar{x}^u}g_{ab}$ then apply the geodesic equation to the metric of Oblate Coordinates and solve it.
I believe it is also possible to calculate the geodesic in Spherical Coordinates and transform your answer to Oblate Coordinates because the solution will be a vector equation. This would obviously be much easier.
Because parametric representations are very similar, the transformation matrices are easy to calculate.
There is also this direct method: http://mathworld.wolfram.com/OblateSpheroidGeodesic.html