Histogram PDF fitting- Bin Variance Poisson or not

So my basic query is just because you've binned and counted the data into a histogram are the error bars on the bins necessarily the Poisson sqrt(N)?

Longer explanation: I have an image of pixel values which represent brightness values. Im binning them into a histogram to get the PDF of the brightness distribution and then trying to fit a predicted model PDF to that. I'm fitting just using chi-square/least squares but if I use sqrt(N) as the error bars then there is basically never a "good" (reduced chi-squared of 3 or 4 for ~200 DOF) even for models we think should be good fits, the error bars seem too small.

We only have one real data image so I tried using one model and generating a bunch (~30,000) simulated images drawing the pixel values from the model and randomizing the positions and making histograms for each one. I then compute the mean of each bin from these simulations and the standard deviation. The standard deviations are definitely larger than sqrt(N). And now if I take the PDF from one of those and compare it the predicted model PDF it gives a good chi squared using the errors from the simulation but again still bad if I use Poisson variance.

But my advisor/boss doesn't agree/understand. He says we're just counting so theres no reason that the error bars should be larger. My idea is that they're larger/not Poisson because its not a totally random process. The number of pixels in a bin is dependent upon the underlying model and how well your particular realization/image samples that underlying distribution.

I'm not a statistics expert and have been doing a lot of reading and just getting a bit confused. If anyone could maybe help clarify or shed some possible light I'd appreciate.