= Solution
All <cohomology> below has rational coefficients. First describe the incidence space geometrically. The image of $g$ is $\mathbb P(W)\cong\mathbb {CP}^1$, inside $\operatorname{Gr}(1,4)\cong\mathbb {CP}^3$. For a line $U\subset W$, choosing a plane $V$ containing $U$ is equivalent to choosing the line $V/U$ in $\mathbb C^4/U$. If $\mathcal L$ is the tautological line bundle on $\mathbb P(W)$, then
$$
F_W\cong\mathbb P\big((\mathbb P(W)\times\mathbb C^4)/\mathcal L\big).
$$
This is the <projective bundle> of a rank-three quotient bundle, with fibre $\mathbb {CP}^2$. Local frames of the quotient bundle give local product charts. In particular $F_W$ is an oriented real six-dimensional <manifold>.
For $V\in S_W\setminus\{W\}$, the intersection $V\cap W$ is exactly a line, so $f^{-1}(V)$ is a single point and the inverse map is $V\mapsto(V\cap W,V)$. The intersection line varies continuously, indeed holomorphically, wherever its dimension is constant: in local <Grassmannian> charts it is the kernel of a matrix of constant rank, whose kernel bases can be chosen locally. Thus $f$ is an isomorphism over the open stratum. At the remaining point,
$$
\boxed{f^{-1}(W)=\mathbb P(W)\cong\mathbb {CP}^1.}
$$
Set $K=Rf_*\mathbb Q_{F_W}$, a <derived direct image of sheaves>. The <proper base change for sheaves> theorem identifies its derived stalks with the <cohomology> complexes of the fibres. Consequently $K$ is the constant rational <sheaf> in degree zero on $S_W\setminus\{W\}$, and
$$
H^a(i_W^*K)=\begin{cases}\mathbb Q,&a=0,2,\\0,&a\ne0,2.\end{cases}
$$
The <cohomology sheaves> are therefore finite-rank <local systems> on the two strata, with no other nonzero degrees. This proves that $K$ is a <constructible complex of sheaves> and verifies normalization. The open stratum has real dimension six, so the isolated point has codimension six. Its middle <perversity> is $\bar m(6)=2$, and the <intersection-complex support and cosupport axioms> require stalk vanishing in degrees above two. The fibre calculation verifies the support axiom, as well as vanishing in negative degrees.
For an oriented real six-dimensional <manifold>, <Verdier duality> gives $D_{F_W}\mathbb Q_{F_W}\cong\mathbb Q_{F_W}[6]$. For a <proper map>, <Verdier duality> commutes with derived direct image. Applying these statements here yields
$$
D_{S_W}K\cong Rf_*D_{F_W}\mathbb Q_{F_W}\cong Rf_*\mathbb Q_{F_W}[6]=K[6],\qquad
\boxed{\Sigma^{-6}D_{S_W}K\cong K.}
$$
The duality identity $i_W^!D_{S_W}K\cong D_{\{W\}}i_W^*K$ converts this into
$$
H^a(i_W^!K)\cong H^{6-a}(i_W^*K)^*.
$$
Here $i_W^!K$ is the <costalk>, and dualization of rational complexes reverses degrees. Thus its only nonzero groups are $\mathbb Q$ in degrees four and six. In particular they vanish for $a\le3$, which is precisely the cosupport axiom at this isolated six-dimensional singularity. The uniqueness characterization by constructibility, normalization, support and cosupport now identifies $K$ with the unshifted <intersection complex> of $S_W$. Taking <hypercohomology>, and using the composition of derived global sections with $Rf_*$, gives
$$
\boxed{IH^*(S_W)\cong\mathbb H^*(S_W,K)\cong H^*(F_W).}
$$
Finally apply the fibration result allowed in the question to $g:F_W\to\mathbb P(W)$. Its base is the simply connected <complex projective line>, its fibre is the <Complex projective plane>, and the projective-bundle charts above verify the fibration hypothesis. Hence, as graded rational <vector spaces>,
$$
\boxed{IH^*(S_W)\cong H^*(\mathbb {CP}^1)\otimes H^*(\mathbb {CP}^2).}
$$
The resulting <Betti numbers> and <Poincaré polynomial> are
$$
\boxed{\dim IH^i(S_W)=\begin{cases}1,&i=0,6,\\2,&i=2,4,\\0,&\text{otherwise},\end{cases}\qquad P_{IH}(t)=1+2t^2+2t^4+t^6.}
$$
This uses $H^*(\mathbb {CP}^r)=\mathbb Q[h]/(h^{r+1})$ with $\deg h=2$. No assertion that an arbitrary projective fibration has the product <cohomology> ring is needed.
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