Yes, you can, by adapting the argument from this answer.
Note that in a bipartite graph $G=(A, B, E)$, the set $B$ is a vertex cover.
Lemma 1. Let $G$ be any $n$-vertex connected bipartite graph containing a vertex cover $B$ such that $\Omega(|B|)$ vertices in $B$ have degree at least three. Then $G$ has a connected dominating set whose complement has size $\Omega(n)$.
Before we give the proof, note that $B$ above exists if and only if the following holds: the number of edges in $G$ such that both endpoints have degree at most two is $O(1)$ times the number of vertices in $G$ of degree three or more.
Proof of Lemma 1. Let $T$ be a spanning tree of $G$ with a maximum number of leaves. Let $n_1$ and $n_2$ be the numbers of leaves and degree-2 vertices in $T$. Let $n^+_3$ be the number of vertices of degree three or more in $T$. Let $n$ be the number of vertices in $G$.
Let $G_2$ be the subgraph of $T$ consisting of the $n_2$ vertices of degree 2 in $T$ and the edges in $T$ between these vertices. (Note that $G_2$ is a subgraph of $T$, and has maximum degree 2, so is a collection of paths.) As shown in that answer, the number of edges in $G_2$ is at least $n - 4n_1$, and the number of vertices in $G_2$ is at least $n-2n_1$.
As $G_2$ has at most $n$ nodes and at least $n-4n_1$ edges, $G_2$ consists of at most $4 n_1$ paths. Each path in $G_2$ contains at most one more vertex from $A$ than from $B$ (as no edge in the path has both endpoints in $A$) so, letting $A$ be the set of vertices not in $B$, we have $|B\cap G_2| \ge |A\cap G_2| - 4n_1$.
As $|B\cap G_2| + |A\cap G_2| = |G_2| \ge n-2n_1$, it follows that $2|B\cap G_2| \ge n - 6n_1$, which implies $|B| \ge n/2 - 3n_1$.
The non-leaf vertices of $T$ form a connected dominating set, whose complement (the leaves) has size $n_1$. So if $n_1 \ge n/9$ we are done.
In the remaining case we have $|B| \ge n/2 - 3n/9 = \Omega(n)$, so $G$ has $\Omega(n)$ vertices of degree three or more, and by Lemma 1 in the previous answer, $G$ has a connected dominating set whose complement has size $\Omega(n)$. $~~~\Box$