Whatever 'high-fallutin' references you come up with, GROUND speed is ALWAYS speed relative to the ground. Nothing else, pure and simple. The OP sought a way of correcting for the circumference of an ellipsoid shell, which is relevant when dealing with travel around an object rather than travel along an object, and the cause of many conundrums and fixes in science and aviation, including such concepts as 'profile rate' which needs to be applied to an inertial attitude reference platform when moving around a body.
To derive either circumferential distance/speed travelled around the shell or convert that to distance relative to the surface of the shell is indeed a problem for the OP (and will also depend on whether the GPS has an altitude function and on which datum that is based), but what he/she refers to is not necessarily GROUNDspeed but some other speed - hence the confusion. As I said earlier, what is sought is to derive 'groundspeed' from 'Tangspeed' or some derivative of Orbital Velocity. (Never did get those postcards....). That can easily be derived by schoolboy mathematics. The moment you begin to 'orbit' just 1mm from the surface of a body the 'speed' link with that surface is broken.
We should think ourselves indeed lucky that our earth is not cuboid - then we'd have some fun..