Such "Krull-Schmidt" type questions have an interpretation in terms of $K_0$ of the category of Chow motives about which very little is known. Some nontrivial tricks how to work with $K_0$ of an additive category are explained in Section 2 of Efimov's paper on L-equivalence arxiv.org/pdf/1707.08997.pdf.

One positive result is Lemma 1 in arxiv.org/pdf/0806.0173.pdf: it basically says that a motive with vanishing Chow groups over all field extensions is zero. However it is not sufficient for the positive answer to your question as Chow groups are infinite dimensional in general.

For the intersection homology one can use the definition using cycles (this is a bit cumbersome), the open-closed exact sequence (removing the vertex of the cone) or the exact sequence involving the resolution of $Y$ (which is a $\mathbb{P}^1$-bundle over $C$).

A cheap way to compute cohomology of the projective cone is to use that in the Grothendieck ring of varieties $[Y] = 1 + \mathbf{L}[C]$ which gives the correct Poincare polynomial.

Lci subvarieties are locally transverse intersections of Cartier divisors, hence one may try to understand the case of a Cartier divisor first.

I think if you take $X = V(xy - z^2) \subset \mathbb{P}^3_{xyzw}$, $D = X \cap V(z)$ a Cartier divisor (pair of lines on $X$), and $Z$ a general line through $[0:0:0:1] = P \in X$, then LHS = 1, and RHS = 2?

Welcome to Mathoverflow! Let $C_1$, $C_2$ be distinct $(-2)$-curves on a smooth projective surface; assume that $[C_1] = [C_2]$ as rational homology classes. Then we get a contradiction $-2 = C_1 \cdot C_2 \ge 0$ (the last inequality holds as the curves don't share common components, so intersect effectively).

@Friedrich: I don't think so. If E is an elliptic curve, then $NS(E \times E)$ will have rank $3$ or $4$ depending on whether $E$ has complex multiplication or not, as the graph of the complex multiplication provides an extra divisor.

Just a comment that if $X = Y \times Z$, then $Pic(X)$ is not in general isomorphic to $Pic(Y) \oplus Pic(Z)$. Indeed, if $Y$ and $Z$ are smooth projective curves of positive genus, then the extra divisors on $Y \times Z$ come from correspondences between $Y$ and $Z$.

Just a comment that since $X$ is Gorenstein, $\omega_X$ is a line bundle hence local Hom to $\mathcal{O}_X$ and to $\omega_X$ are essentially the same.

Actually, I think if you assume the statement for stable $\mathcal{E}$, then the one for general $\mathcal{E}$ may follow by the following induction argument: consider the HN filtration on $\mathcal{E}$, and take its limit to the special fiber.

If you restrict to only semistable bundles, then points of the coarse moduli space are S-equivalence classes which is kind of a filtration like you ask?

I actually wonder if the J-map alone can prove the easier "Exit quiz" claim (only isotrivial fibrations over $\mathbb{P}^1 \setminus \{0,\infty\})$. Clearly there are infinitely many (families of) nonconstant maps $\mathbb{P}^1 \setminus \{0,\infty\}) \to \mathbb{A}^1$ but somehow they are not realized by elliptic fibrations.

Why is $S = J^{-1}(\{0,1,\infty\})$? I would think that $J(S) = \{\infty\}$?