Analytic Hull White model with correlated stochastic processes

I am trying to price a path dependent option which uses two underlyings (a stock index and an interest rate index). I am using Hull White model for interest rate modelling and local vol for stock index modelling. $$\begin{eqnarray} dr(t) &=& (\theta(t) - kr(t))dt + \sigma_r(t)dw_r(t)\\ \end{eqnarray}$$

$$\begin{eqnarray} ds(t) &=& r'(t)sdt + \sigma_s(t,s)sdw_s(t)\\ \end{eqnarray}$$ Further, assume that $$ = \rho dt$$ and that $$r'$$ is deterministic. I need the samples for $$s$$ and $$r$$(and stochastic discount factors) only at certain times $$t_1, t_2...t_n$$. In case we did not have correlated processes, I could have calculated mean and variance of $$r$$ and discount factors analytically (and then creating the corresponding samples). Can we still solve this by calculating the mean and variance for $$r$$ and discount factors analytically and simulate $$s_t$$ numerically?

Yes, in the case of path-dependent option with two underlying assets, you can approach the problem by calculating the mean and variance for the interest rate $$r(t)$$ and corresponding discount factor analytically, while simulating the stock price $$S(t)$$ numerically. This will still work with correlated process since we have the correlation coefficient between the Brownian motions.
You can solve the Hull-White SDE analytically to find the mean and variance of the interest rate at future times $$t_1,t_2,...,t_n$$, then simulate the interest rate path.
For the local volatility model for the stock price, $$r'(t)$$ is deterministic, and $$\sigma_s(t,s)$$ is the local volatility model which is dependent on time and the stock price, you can typically simulate this process numerically. You can look into Euler–Maruyama method.
Then when simulating the paths for $$S(t)$$ and $$r(t)$$ just make sure you account for the correlation. Using the simulated paths for $$r(t)$$, you can calculate the discount factor for the respective time $$t_1,t_2,...,t_n$$.