That paper states, on page 1038, “Given the simulated choice set, I compute choice probabilities for each individual for each product and construct an importance sampler to smooth the simulated choice probabilities.” With this, the moment conditions are differentiable, so the standard extremum estimation GMM methods can be applied to get standard errors. This smoothing idea is one that started with McFadden, as far as I know. If you implement smoothing, then I think that automatic differentiation will work just fine to get the Hessian.