Assume the correlation $$\rho$$ is constant.
(2-equal-weighted underlying for demonstration purposes) $$\sigma_{basket}^2(S^a_{t},S^b_{t},t) = (\frac{1}{2}\sigma_{a}(S^a_{t},t))^2 + 2\rho(\frac{1}{2}\sigma_{a}(S^a_{t},t))(\frac{1}{2}\sigma_{b}(S^b_{t},t))+ (\frac{1}{2}\sigma_{b}(S^b_{t},t))^2$$
Your equation holds if you replace $$\sigma$$ with $$\text{var}$$. But if you use it as such, even in the BS case ($$\sigma(S,t) = \sigma$$) then you have a problem, because you are manipulating a lognormal volatility and the (weighted) sum of lognormals is not lognormal. Now nothing prevents you from simulating the future price processes of assets $$a$$ and $$b$$ tying them by a Gaussian copula with parameter $$\rho$$, pricing vanillas on the basket and deriving an IVS for the basket. Strip this one to get the basket's LV.