In this paper we present a data analysis approach applicable to the potential saddle-point fly-by mission extension of LISA Pathfinder (LPF). At the peak of its sensitivity, LPF will sample the gravitational field in our Solar System with a precision of several $\text{fm/s}^2/\sqrt{\text{Hz}}$ at frequencies around $1\,\text{mHz}$. Such an accurate accelerometer will allow us to test alternative theories of gravity that predict deviations from Newtonian dynamics in the non-relativistic limit. As an example, we consider the case of the Tensor-Vector-Scalar theory of gravity and calculate, within the non-relativistic limit of this theory, the signals that anomalous tidal stresses generate in LPF. We study the parameter space of these signals and divide it into two subgroups, one related to the mission parameters and the other to the theory parameters that are determined by the gravity model. We investigate how the mission parameters affect the signal detectability concluding that these parameters can be determined with the sufficient precision from the navigation of the spacecraft and fixed during our analysis. Further, we apply Bayesian parameter estimation and determine the accuracy to which the gravity theory parameters may be inferred. We evaluate the portion of parameter space that may be eliminated in case of no signal detection and estimate the detectability of signals as a function of parameter space location. We also perform a first investigation of non-Gaussian "noise-glitches" that may occur in the data. The analysis we develop is universal and may be applied to anomalous tidal stress induced signals predicted by any theory of gravity.