Eliminating inertia effect from the acceleration signal
It sounds like you want to remove the frequency component at about 2.5Hz in the plot, but leave the rest of the plot as untouched as possible.

The usual first approach to that problem is to just use a notch filter:

$$
H(z) = K \displaystyle\frac{1 - 2 \cos \omega_0 z^{-1} + z^{-2}}{1 - 2r \cos\omega_0 z^{-1} + r^2 z^{-2}}
$$

where $\omega_0$ is the (normalized) frequency you want to remove and $r \lt 1$.

There are more complex approaches that use, for example, Kalman filtering... but that might require more assumptions about the dynamics of your system that you are prepared to make given your "black box" comments.