Observe that the assumptions imply that $E_2|_{X \setminus Z}$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p_1 + p_3$. Thus the statement makes sense. The proof of this statement is tricky because the operator $\eta_\mathcal{I}$ from More on Flatness, Section \ref{flat-section-eta} used in the construction of localized Chern classes doesn't transform distinguished triangles into distinguished triangles. \medskip\noindent Let $\varphi^\bullet : \mathcal{E}_3^\bullet[-1] \to \mathcal{E}_1^\bullet$ be a map of bounded complexes of finite locally free $\mathcal{O}_X$-modules representing $E_3[-1] \to E_1$ which exists by assumption. Consider the scheme $X' = \mathbf{A}^1 \times X$ with projection $g : X' \to X$. Let $Z' = g^{-1}(Z) = \mathbf{A}^1 \times Z$. Denote $t$ the coordinate on $\mathbf{A}^1$. Consider the cone $\mathcal{C}^\bullet$ of the map of complexes $$ t g^*\varphi^\bullet : g^*\mathcal{E}_3^\bullet[-1] \longrightarrow g^*\mathcal{E}_1^\bullet $$ over $X'$. We obtain a distinguished triangle $$ g^*\mathcal{E}_1^\bullet \to \mathcal{C}^\bullet \to g^*\mathcal{E}_3^\bullet \to g^*\mathcal{E}_1^\bullet[1] $$ where the first three terms form a termwise split short exact sequence of complexes. Clearly $\mathcal{C}^\bullet$ is a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules whose restriction to $X' \setminus Z'$ is isomorphic to a finite locally free $\mathcal{O}_{X' \setminus Z'}$-module of rank $< p_1 + p_3$ placed in degree $0$. Thus we have the localized Chern classes $$ c_p(Z' \to X', \mathcal{C}^\bullet) \in A^p(Z' \to X') $$ for $p \geq p_1 + p_3$. For any $\alpha \in \CH_k(X)$ consider $$ c_p(Z' \to X', \mathcal{C}^\bullet) \cap g^*\alpha \in \CH_{k + 1 - p}(\mathbf{A}^1 \times X) $$ If we restrict to $t = 0$, then the map $t g^*\varphi^\bullet$ restricts to zero and $\mathcal{C}^\bullet|_{t = 0}$ is the direct sum of $\mathcal{E}_1^\bullet$ and $\mathcal{E}_3^\bullet$. By compatibility of localized Chern classes with base change (Lemma \ref{lemma-base-change-loc-chern}) we conclude that $$ i_0^* \circ c^{(p_1 + p_3)}(Z' \to X', \mathcal{C}^\bullet) \circ g^* = c^{(p_1 + p_2)}(Z \to X, E_1 \oplus E_3) $$ in $A^{(p_1 + p_3)}(Z \to X)$. On the other hand, if we restrict to $t = 1$, then the map $t g^*\varphi^\bullet$ restricts to $\varphi$ and $\mathcal{C}^\bullet|_{t = 1}$ is a bounded complex of finite locally free modules representing $E_2$. We conclude that $$ i_1^* \circ c^{(p_1 + p_3)}(Z' \to X', \mathcal{C}^\bullet) \circ g^* = c^{(p_1 + p_2)}(Z \to X, E_2) $$ in $A^{(p_1 + p_3)}(Z \to X)$. Since $i_0^* = i_1^*$ by definition of rational equivalence (more precisely this follows from the formulae in Lemma \ref{lemma-linebundle-formulae}) we conclude that $$ c^{(p_1 + p_2)}(Z \to X, E_2) = c^{(p_1 + p_2)}(Z \to X, E_1 \oplus E_3) $$ This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume $E_2 = E_1 \oplus E_3$ and $E_i$ corresponds to $\mathcal{E}_i^\bullet$ for $i = 1, 3$ as above. Set $\mathcal{E}_2^\bullet = \mathcal{E}_1^\bullet \oplus \mathcal{E}_3^\bullet$. For $i = 1, 2, 3$ let $$ b_i : W_i \to \mathbf{P}^1_X \quad\text{and}\quad \mathcal{Q}_i^\bullet $$ be the blowing up and complex of $\mathcal{O}_{W_i}$-modules constructed in More on Flatness, Section \ref{flat-section-blowup-complexes-III}. Let $T_i \subset W_{i, \infty}$ be the closed subscheme whose existence is averted in More on Flatness, Lemma \ref{flat-lemma-graph-construction}. Let $T'_i \subset T_i$ be the open and closed subscheme such that $\mathcal{Q}_i^\bullet|_{T'_i}$ is isomorphic to a finite locally free sheaf of rank $< p_i$ in degree $0$. By definition $$ c_p(Z \to X, E_i) = c'_p(\mathcal{Q}_i^\bullet) $$ where the right hand side is the bivariant class constructed in Lemma \ref{lemma-localized-chern-pre} using $W_i, b_i, \mathcal{Q}_i^\bullet, T'_i$. By Divisors,  Lemma \ref{divisors-lemma-blowing-up-two-ideals} we can choose a commutative diagram $$ \xymatrix{ W \ar[dd]^{g_1} \ar[rd]^{g_2} \ar[rr]_{g_3} & & W_3 \ar[dd]^{b_3} \\ & W_2 \ar[rd]^{b_2} \\ W_1 \ar[rr]^{b_1} & & \mathbf{P}^1_X } $$ where all morphisms are blowing ups which are isomorphisms over $\mathbf{A}^1_X$. \medskip\noindent By Lemma \ref{lemma-localized-chern-pre-independent} we may use $W$, $b = b_i \circ g_i$, $g_i^*\mathcal{Q}_i^\bullet$, and $g_i^{-1}(T'_i)$ to construct $c_p(Z \to X, E_i)$. The same lemma also tells us that we may replace $g_i^{-1}(T'_i)$ with $T = g_1^{-1}(T'_1) \cap g_2^{-1}(T'_2) \cap g_3^{-1}(T'_3)$ because this closed subscheme still contains all points of $W_\infty$ lying over $X \setminus Z$. Hence we may use $T$ for each of the three constructions. By More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-eta-pull} applied to the morphisms $g_i : W \to W_i$ we find that $$ g_i^*\mathcal{Q}_i^\bullet = \eta_\mathcal{I}b^*p^*\mathcal{E}_i^\bullet $$ where $\mathcal{I}$ is the invertible ideal sheaf of the effective Cartier divisor $W_\infty$ and $p :\mathbf{P}^1_X \to X$ is the projection morphism. Since the functor $\eta_\mathcal{I}$ visibly commutes with direct sums, we see that $\mathcal{Q}_2^\bullet = \mathcal{Q}_1^\bullet \oplus \mathcal{Q}_3^\bullet$. Thus the desired equality follows from Lemma \ref{lemma-localized-chern-pre-sum-c}.