The Gravity Advanced Package is an instrument composed of an electrostatic accelerometer called MicroSTAR and a rotating platform called Bias Rejection System. It aims at measuring with no bias the non-gravitational acceleration of a spacecraft. It is envisioned to be embarked on an interplanetary spacecraft as a tool to test the laws of gravitation. MicroSTAR is based on Onera's experience and inherits in~orbit technology. The addition of the rotating platform is a technological upgrade which allows using an electrostatic accelerometer to make measurements at low frequencies with no bias. To do so, the Bias Rejection System rotates MicroSTAR such that the signal of interest is separated from the bias of the instrument in the frequency domain. Making these unbiased low-frequency measurements requires post-processing the data. The signal processing technique developed for this purpose is the focus of this article. It allows giving the conditions under which the bias is completely removed from the signal of interest. And the precision of the unbiased measurements can be fully characterized: given the characteristics of the subsystems, it is possible to reach a precision of 1 pm s$^{-2}$ on the non-gravitational acceleration for an integration time of 3 h.