Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V_0 \subseteq V_1 \subseteq V_2$. However, a variant on your construction does have a positive answer, and this is closely related to Massey products. Recall first that if $(A_\bullet,d)$ is a (homologically graded) dg-algebra then an $n$-fold Massey system on $A$ is a collection of elements $a_{i,j} \in A_{-1}$ for $0 \leq i < j \leq n$ with $(i,j) \neq (0,n)$ satisfying the relation $\partial a_{i,j} = \sum_{k=i+1}^{j-1} a_{i,k}a_{k,j}$ (in particular, $\partial a_{i,i+1} =0$). In this case, the element $b := \sum_{k=1}^{n-1}a_{0,k}a_{k,n} \in A_{-2}$ is closed and its class $[b] \in H_{-2}(A)$ is called the value of the Massey system $\{a_{i,j}\}$. Classically this value is often considered as the $n$-fold Massey product of the classes $[a_{i,i+1}] \in H_{-1}(A)$, although one should remember that this value depends on the choice of a Massey system, and not just on the $[a_{i,i+1}]$. A trivialization of a Massey system $\{a_{i,j}\}$ is an element $a_{0,n} \in A_{-1}$ satisfying $\partial a_{0,n} = \sum_{k=1}^{n-1}a_{0,k}a_{k,n}$, i.e., witnessing the vanishing of the associated $n$-fold Massey product.
The main observation (which I believe, under a somewhat different formulation, is essentially due to Dwyer), is that the data $(\{a_{i,j}\},a_{0,n})$ of an $n$-fold Massey system equipped with a trivialization classifies (right) $A$-modules $M$ equipped with an ascending filtration
$$0 = M_{-1} \to M_0 \to M_1 \to ...\to M_n=M $$
such that each homotopy cofiber $M_{i}/M_{i-1}$ is (identified with) $A$. For example, suppose that $n=1$. Then the data of a trivialized Massey system is just the data of a closed element $a_{0,1} \in A_{-1}$, which can be considered as a map of $A$-modules $a_{0,1}: A \to A[1]$. One then defines $M$ to be the homotopy fiber of this map, so that $M$ sits in a homotpy fiber sequence $A \to M \to A$. In particular, $M$ is equipped with a $2$-step filtration
$$ 0 \to A \to M $$
whose successive cofibers are $A$. Now consider the case $n=2$. Then we are given closed elements $a_{0,1},a_{1,2} \in A_{-1}$ and a trivialization $a_{0,2} \in A_{-1}$ satisfying $\partial a_{0,2} = a_{0,1}a_{1,2}$. We may then consider $a_{0,1}$ and $a_{1,2}$ as $A$-module maps $a_{0,1}: A[1] \to A[2]$ and $a_{1,2}: A[0] \to A[1]$, and consider $a_{0,2}$ as a null-homotopy of their composition $a_{0,1} \circ a_{1,2}$. Let $M_{0,1}$ and $M_{1,2}$ be the homotopy fibers of $a_{0,1}$ and $a_{1,2}$ respectively. Then the null-homotopy $a_{0,2}$ determines a lift of $a_{1,2}: A \to A[1]$ to a map $\tilde{a}_{1,2}: A \to M_{0,1}$. Let $M_{0,2}$ be the homotopy fiber of $\tilde{a}_{1,2}$. Then $M_{0,2}$ is equipped with a $3$-step filtration
$$ 0 \to A \to M_{0,1}[-1] \to M_{0,2} $$
whose successive cofibers are $A$.
This idea can be generalized to describe filtered $A$-modules with other types of successive cofibers. Indeed, if $B_0,...,B_n$ are fixed $A$-modules, then we can define the notion of a Massey system on $B_0,...,B_n$ to be a collection of elements $a_{i,j} \in \underline{\rm Hom}_{-1}(B_j,B_i)$ (where $\underline{\rm Hom}_\bullet$ means the mapping chain-complex of $A$-modules) for $(i,j) \neq (0,n)$ such that $\partial a_{i,j} = \sum_{k=i+1}^{j-1} a_{i,k}\circ a_{k,j}$, and define a trivialization of $\{a_{i,j}\}$ to be an element $a_{0,n} \in \underline{\rm Hom}_{-1}(B_n,B_0)$ such that $\partial a_{0,n} = \sum_{k=1}^{n-1}a_{0,k} \circ a_{k,n}$. Then the data of a Massey system on $B_0,...,B_n$ equipped with a trivialization then classifies $A$-modules equipped with an $(n+1)$-step filtration whose successive cofibers are $B_0,...,B_n$. Let's call such objects $(B_0,...,B_n)$-filtered $A$-modules. You can also ask, what about Massey systems $\{a_{i,j}\}$ not-necessarily trivialized? Well, these will classify triples $(M_{0,n-1},M_{1,n},f)$ where $M_{0,n-1}$ is a $(B_0,...,B_{n-1})$-filtered $A$-module, $M_{1,n}$ is a $(B_1,...,B_n)$-filtered $A$-module and $f$ is an equivalence of $(B_1,...,B_{n-1})$-filtered $A$-modules between ${\rm cofib}(B_0 \to M_{0,n-1})$ and ${\rm fib}(M_{1,n}\to B_n)$. The associated $n$-fold Massey product $[\sum_{k=1}^{n-1}a_{0,k} \circ a_{k,n}] \in H_{-2}(\underline{\rm Hom}(B_n,B_0))$ (which can be considered as a homotopy class of maps $B_n \to B_0[-2]$) is then the obstruction to finding a $(B_0,...,B_n)$-filtered $A$-module $M_{0,n}$ such that ${\rm fib}(M_{0,n}\to B_n) \simeq M_{0,n-1}$ and ${\rm cofib}(B_0 \to M_{0,n}) \simeq M_{1,n}$.
Of course, what can be done for modules can usually be done for an arbitrary stable $\infty$-category. Indeed, if $B_0,B_1,...,B_n$ are fixed objects in a stable $\infty$-category ${\cal C}$ then we can classify $(B_0,B_1,...,B_n)$-filtered objects by suitable Massey systems $\{a_{i,j}\}$, where the $a_{i,i+1}$ are maps of the form $a_{i,i+1}: B_{i+1}[n-i-1] \to B_{i}[n-i]$, $a_{i,i+2}$ is a null-homotopy of the composition $a_{i,i+1} \circ a_{i+1,i+2}$, etc. The case you are asking about is the case of a $3$-step filtration with successive cofibers $A,B,C \in {\cal C}$. In this case $(A,B,C)$-filtered objects are classified by the data $(\alpha',\beta,h)$ consisting of maps $\alpha': B[1] \to A[2]$ and $\beta: C \to B[1]$ and a null-homotopy $h$ of $\alpha' \circ \beta$. Here $\alpha'$ is a shift of your $\alpha$. The null-homotopy $h$ is actually determined by either a lift of $\beta$ to $\gamma: C \to D[1]:= {\rm fib}(\alpha')$ or an extension of $\alpha'$ to $\delta': E[1] := {\rm cofib}(\beta) \to A[2]$. This is essentially what you did with your $\gamma$ and $\delta$. However, to get the null-homotopy $h$ it is not sufficient to remember just $\gamma$ (or $\delta$), one also need to keep the homotopy relating the composition $C \stackrel{\gamma}{\to} D[1] \to B[1]$ to $\beta$. Of course if you do this for both $\gamma$ and $\delta$ you end up encoding this null-homotopy twice, so you will need to add a homotopy relating the two choices. Alternatively, just drop one of $\gamma,\delta$ (and add the missing homotopy instead).
