= Solution
For an open subset $U\subseteq B$, let $E_U=\pi^{-1}(U)$ and define the degree-preserving module map
$$
\Phi_U:\bigoplus_{j=1}^vH^{*-k_j}(U;\mathbb Q)\longrightarrow H^*(E_U;\mathbb Q),\qquad
(b_j)_j\longmapsto\sum_j\pi^*b_j\smile c_j|_{E_U}.
$$
Using the prescribed fiber basis identifies its domain with the tensor product in the question. We prove that $\Phi_B$ is an <isomorphism>, giving the <Leray-Hirsch theorem> in this setting.
First suppose $U$ is a trivializing patch. The <cohomological Künneth theorem over a field> identifies $H^*(U\times F;\mathbb Q)$ with the finite free graded $H^*(U;\mathbb Q)$-module on a homogeneous basis of $H^*(F;\mathbb Q)$. The required finiteness is supplied by the finite list $c_1,\ldots,c_v$ spanning the fiber's total cohomology. Express the restrictions of the $c_j$ in this product basis. A coefficient multiplying a fiber basis element of degree $k_i$ belongs to $H^{k_j-k_i}(U;\mathbb Q)$; thus no coefficient can increase fiber degree. The degree-zero blocks are invertible on every path component, because restriction at every point gives a basis. Their inverses are again degree-zero cohomology classes.
After inverting those blocks, the remaining change-of-basis map is $I+N$, where $N$ strictly lowers fiber degree. There are only finitely many fiber degrees, so $N$ is <nilpotent> and $(I+N)^{-1}=I-N+N^2-\cdots$ is a finite sum. This proves the <local basis criterion for Leray-Hirsch classes> and therefore the isomorphism on every trivializing patch, without requiring the patch to be contractible.
Now use a finite trivializing open cover $U_1,\ldots,U_r$, supplied by <compactness> of $B$. Induct on $r$. Put $U=U_1\cup\cdots\cup U_{r-1}$ and $V=U_r$. The intersection $U\cap V$ is covered by the at most $r-1$ trivializing open sets $U_i\cap V$, so the induction hypothesis applies to $U$, $V$ and $U\cap V$.
Take the direct sum of shifted <Mayer–Vietoris sequences> for the base and compare it with the sequence for $E_U\cup E_V$. Restriction commutes with multiplication by each global <cohomology class> $c_j$. The connecting homomorphism also commutes: multiply the cochain representatives on the right by the restriction of a global cocycle representing $c_j$, and use $d(b\smile c_j)=db\smile c_j$. Consequently the maps $\Phi$ form a morphism of exact sequences. The <Five lemma> makes $\Phi_{U\cup V}$ an isomorphism, completing the <finite-cover proof of the Leray-Hirsch theorem>.
This is an additive graded module isomorphism. It does not assert that the fiber classes have the same multiplicative relations in the total space.
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