A very nice project! The poor fit to a Poisson distribution is not likely the result of mixing particles: each might have its own rate, but the sum of random events with different rates is just a Poisson with the total rate. You don't really explain whether you are binning over space, time or both, but later you show that the rates vary over time (during the day), so if you are binning over time that would explain why you see the wider spread of observed data values: high value come from the middle of the day, when the rate is higher than assumed, and low from the night, when it is less. There is a way to model this: you specify a Poisson distribution where lambda comes from another distribution, which you can get by fitting your data on the temporal distribution.
