It actually isn't true that the complexity of $\theta$ could be meaningfully bounded in the term of complexities of $\varphi$ and $\psi$.

Let us fix an arbitrary recursive ordinal $\alpha$. Below I sketch a construction of infinitary $\Pi_n$ formulas $\varphi,\psi$, for some finite $n$ such that there are no $\Pi_\alpha$ interpolant for them.

We fix some $\Delta^1_1$ in the standard model $\mathbb{N}$ of $\mathsf{PA}$ property $F(X)$ of sets $X\subseteq \mathbb{N}$ that it is not $\boldsymbol\Pi_\alpha$. For example, $F(X)$ could be the property of $X$ to encode an isomorphic copy of $\omega^{\alpha+1}$. Next we fix first-order arithmetical formulas $\varphi'(X,Y)$ and $\psi'(X,Y)$ depending on free unary predicates such that $$F(X)\iff \mathbb{N}\models_2 \exists Y\; \varphi'(X,Y)\iff \mathbb{N}\models_2 \forall Z\; \psi'(X,Z).$$

We put $$\varphi(X,Y) \mathrel{:{=}} \bigwedge \mathsf{Q} \land \forall x \bigvee\limits_{n<\omega} x=S^n(0)\to \varphi'(X,Y)\text{ and}$$ $$\psi(X,Z) \mathrel{:{=}} \bigwedge \mathsf{Q} \land \forall x \bigvee\limits_{n<\omega} x=S^n(0)\to \psi'(X,Z),$$ where $\bigwedge \mathbf{Q}$ is the conjuction of the axioms of Robinson's arithmetic $\mathsf{Q}$. Observe that any interpolant $\theta(X)$ for this pair of $\varphi(X,Y)$ and $\psi(X,Z)$ should express the property $F(X)$ in $\mathbb{N}$. Thus $\theta(X)$ couldn't be a $\Pi_\alpha$ infinitary formula.

It actually isn't true that the complexity of $\theta$ could be meaningfully bounded in the term of complexities of $\varphi$ and $\psi$.

Let us fix an arbitrary recursive ordinal $\alpha$. Below I sketch a construction of infinitary $\Pi_n$ formulas $\varphi,\psi$, for some finite $n$ such that there are no $\Pi_\alpha$ interpolant for them.

Consider the standard model $\mathbb{N}$ of $\mathsf{PA}$. We fix some $\Delta^1_1$-property $F(X)$ of sets $X\subseteq \mathbb{N}$ that it is not $\boldsymbol\Pi_\alpha$. For example, $F(X)$ could be the property of $X$ to encode an isomorphic copy of $\omega^{\alpha+1}$. Next we fix first-order arithmetical formulas $\varphi'(X,Y)$ and $\psi'(X,Y)$ depending on free unary predicates such that $$F(X)\iff \mathbb{N}\models_2 \exists Y\; \varphi'(X,Y)\iff \mathbb{N}\models_2 \forall Z\; \psi'(X,Z).$$

We put $$\varphi(X,Y) \mathrel{:{=}} \bigwedge \mathsf{Q} \land \forall x \bigvee\limits_{n<\omega} x=S^n(0)\to \varphi'(X,Y)\text{ and}$$ $$\psi(X,Z) \mathrel{:{=}} \bigwedge \mathsf{Q} \land \forall x \bigvee\limits_{n<\omega} x=S^n(0)\to \psi'(X,Z),$$ where $\bigwedge \mathbf{Q}$ is the conjuction of the axioms of Robinson's arithmetic $\mathsf{Q}$. Observe that any interpolant $\theta(X)$ for this pair of $\varphi(X,Y)$ and $\psi(X,Z)$ should express the property $F(X)$ in $\mathbb{N}$. Thus $\theta(X)$ couldn't be a $\Pi_\alpha$ infinitary formula.