We assume that the short interest rate $r_t$ follows the Hull-White model, that is, the short rate $r$ and the stock price $S$ satisfies a system of SDEs of the form
\begin{align*}
dr_t &= \lambda(\theta_t -r_t)dt + \sigma_0 dW_t,\\
dS_t &= S_t\Big[r_t dt + \sigma \Big(\rho dW_t + \sqrt{1-\rho^2} dB_t\Big)\Big],
\end{align*}
where $\lambda$, $\sigma_0$, $\sigma$, and $\rho$ are constants, and $\{W_t \mid t>0\}$ and $\{B_t \mid t>0\}$ are independent standard Brownian motions.

Note that,
\begin{align*}
&\ E\bigg(\exp\Big(-\int_0^T r_t dt \Big) (S_T-K)^+\bigg) \\
=& \  E\bigg(e^{-\bar{r}T} \Big(S_0e^{\bar{r}T -\frac{1}{2}\sigma^2 T - \sigma (\rho W_T + \sqrt{1-\rho^2}B_T)} -K\Big)^+ \bigg)\\
=& \ E\Bigg(E\bigg(e^{-\bar{r}T} \Big(S_0e^{\bar{r}T -\frac{1}{2}\sigma^2 T + \sigma (\rho W_T + \sqrt{1-\rho^2}B_T)} -K\Big)^+ \mid r_s, 0<s \leq T\bigg)\Bigg)\\
=& \ E\Big(F(S_0,K,\bar{r},T,\sigma, W_T) \mid r_s, 0<s \leq T\Big),
\end{align*}
for a certain function $F$. Note the random variable $W_T$ in the formula.

If $\rho=0$, that is, $S$ and $r$ are independent, then
\begin{align*}
&\ E\bigg(\exp\Big(-\int_0^T r_t dt \Big) (S_T-K)^+\bigg) \\
=& \ E\Bigg(E\bigg(e^{-\bar{r}T} \Big(S_0e^{\bar{r}T -\frac{1}{2}\sigma^2 T + \sigma B_T} -K\Big)^+ \mid r_s, 0<s \leq T\bigg)\Bigg)\\
=&\ E\Big(BS(S_0,K,\bar{r},T,\sigma) \mid r_s, 0<s \leq T \Big).
\end{align*}
That is, the formula provided in the question holds if the stock price and the interest rate are independent. In this case, $\bar{r}$ can be approximated by a Riemann sum.