Even though the underlying posterior distributions are of different type, the results will be the same if the rate of the Poisson process is sufficiently small and weakly informative and consistent priors are used.
The underlying distribution can be highly skewed, even if the total number of samples in the Monte Carlo simultion is very large. This is why the Normal approximation often performs poorly in practice.
The distribution quantifying the uncertainty about the probability of failure can be highly skewed, even for a large number of samples. The coefficient of variation is easier to interprete for symmetric distributions.
For a given probability of failure, the variance and coefficient of variation of the Monte Carlo estimate can be evaluated analytically. From this, the total number of samples required to maintain a target coefficient of variation can be deduced.