Calculating Median from Fitted Continuous Distributions for Data Analysis

I would like an option to calculate the following metric for a discrete distribution (e.g. number of errors per user per day): fit a common continuous distribution (poisson in my case) to the data and then output the median of that continuous distribution, rather than looking at the median of the discrete distro. My use case is I often have these discrete distributions that are very concentrated near very small values (like 0 or 1 or 2) but have ridiculously long tails (like in the thousands or tens of thousands). The median of this distribution will be 1, and if the distribution shifts to the right slightly the median will still be 1 unless the distribution shifts quite significantly to the right (e.g. the number of errors per user would have to double). So if I'm looking at the median, I lose a ton of precision. But if I'm looking at the average, because of the heavy skew, an overall shift to the right might not even show up if there's even a small shift to the left in the 99th percentile, i.e. the signal gets hidden by noise.