There is an interpretation of your inequality using simplicial complex and $h$-vector. Namely, let $\Delta$ be the set of all $P-S$ with $S$ in your set. Then $\Delta$ is a collection closed under inclusion, hence a simplicial complex.

The invariant you are interested in is, up to sign, the alternating sum of $f_i$: the number of $i$-dim faces of $\Delta$. Then, by a well-known formula, it is equal to $|h_d|$, where $h_i$ forms the $h$-vector of $\Delta$, and $d-1$ is the dimension of $\Delta$.

The right hand side of the inequality you are interested in is the number of facets (maximal elements) of $\Delta$. As people have pointed out, in general the inequality you want does not hold.

However, I would like to point out that it is likely to hold under certain extra topological/homological assumptions. For instance, if $\Delta$ is Cohen-Macaulay, then all the $h_i$ are non-negative, and the number of facets is $f_{d-1}=\sum_{i\geq 0} h_i\geq h_d$, which is what you need. In fact, as $h_0=1$ and $h_1=n-d$ where $n=|P|$, you get something a little stronger.

One can prove other inequalities for $h_i$ under weaker conditions. For example, if $\Delta$ satisfies Serre's conditions $(S_{r})$, one still have non-negativity of $h_{\leq r}$, a result first proved by Murai-Terai. I discussed some of them in a recent talk (but it is perhaps a bit algebraic).