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.
EDIT
Let $Q$ denote the risk-neutral measure and $Q^T$ denote the $T$-forward measure. Note that, the forward price $F(t, T)$ has the form \begin{align*} F(t, T) &= E_{Q^T}(S_T \mid \mathcal{F}_t), \end{align*} which is a martingale under the $T$-forward measure $Q^T$ and satisfies and SDE of the form \begin{align*} dF(t, T) = F(t, T) \sigma d\widehat{W}_t. \end{align*} Note that \begin{align*} \frac{dQ^{T}}{dQ}\big|_t = \frac{P(t, T)}{P(0, T)B_t}. \end{align*} Then \begin{align*} E_Q\left(\frac{(S_T-K)^+}{B_T}\right) &= E_Q\left(\frac{(F(T, T)-K)^+}{B_T}\right)\\ &=E_{Q^T}\left(\frac{(F(T, T)-K)^+}{B_T} \left(\frac{dQ^{T}}{dQ}\big|_T\right)^{-1} \right)\\ &=P(0, T)E_{Q^T}\left((F(T, T)-K)^+\right)\\ &=P(0, T)\big[F(0, T)N(d_1) - KN(d_2) \big],\tag{1} \end{align*} where $d_1 = \frac{\ln F(0, T)/K + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}$ and $d_2 = d_1 - \sigma \sqrt{T}$. Note that, in Formula $(1)$, the modeling for the short rate $r_t$ is not needed, while we only need the $T$-year zero rate $R_T$, which can be obtained from a given yield curve.