It depends what you want. In the linked post, they are talking about computing the expected value of Y, given the model. In that case the closed form solution comes from the mean of the log-normal distribution. Since you have samples, you can skip the theory and just compute it as the mean of your samples.
On the other hand, if you just want to know the value of contrib in levels, you can just exp it. The expected value of exp(contrib) should be different from the expected value of y_obs due to Jensen’s Inequality, which you could verify by comparison.