To do the proper SR calculation, though, you have to compare to an observer at rest. Even though we want to think of ourselves as being in an inertial frame, we are not.
Our reference clocks are referenced to be on the geoid - the idealized average sea-level. The earth is an oblate spheroid - there is an equitorial bulge and flattening at the poles - and one must also account for gravitational redshifts. If you solve for the equipotential surface of the geoid, you find that the kinetic term (time dilation) cancels the deformation term (gravitational redshift) so that a clock on the geoid always runs at the same rate, regardless of rotation speed. The rotational velocity is not "disregarded."
GPS satellites are in inclined orbits at about 55 degrees, not equatorial orbits, so your relative velocity calculation is flawed anyway, as velocity is a vector.