This is false for all $a<\frac{1}{3}$.
Claim. For all $\ell \in \mathbb{N}$ exists a bipartite graph $G$ on $3\ell+1$ vertices such that $\delta(G)=\ell$, but $\kappa(G) < \delta(G)$.
Proof. Let $G_1$ and $G_2$ be copies of $K_{\ell, \ell}$ with bipartitions $(A_1,B_1)$ and $(A_2,B_2)$, respectively. Let $X_1 \subseteq A_1$ and $X_2 \subseteq A_2$ both be of size $k$, and let $G$ be the graph obtained from $G_1$ and $G_2$ by identifying $X_1$ and $X_2$. Note that $G$ is a bipartite graph with $4\ell-k$ vertices, minimum degree $\ell$, and vertex-connectivity $k$. Setting $k=\ell-1$ proves the claim.