The MUonE experiment is designed to extract the hadronic contribution to the electromagnetic coupling in the spacelike region $\Delta {\alpha }_{\mathrm{had}}\left(t\right)$ from elastic $e\mu$ scattering. The leading-order hadronic vacuum polarization contribution to the muon $g-2$, $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$, can then be obtained from a weighted integral over $\Delta {\alpha }_{\mathrm{had}}\left(t\right)$. This, however, requires knowledge of $\Delta {\alpha }_{\mathrm{had}}\left(t\right)$ in the whole domain of integration, which cannot be achieved by experiment. In this work, we propose to use Padé and D-Log Padé approximants as a systematic and model-independent method to fit and reliably extrapolate the future MUonE experimental data, extracting $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$ with a conservative but competitive uncertainty, using no or very limited external information. The method relies on fundamental analytic properties of the two-point correlator underlying $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$ and provides lower and upper bounds for the result for $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$. We demonstrate the reliability of the method using toy datasets generated from a model for $\Delta {\alpha }_{\mathrm{had}}\left(t\right)$ reflecting the expected statistics of the MUonE experiment.