How would I fit a distribution to this image noise?
Since you say your distribution is discrete, if your samples are independent, you could try a multinomial distribution as a model
https://en.m.wikipedia.org/wiki/Multinomial_distribution

You can use either maximum likelihood or a Bayesian estimator.  

You could also take the mean and standard deviation and use a Gaussian model with rounding.  A goodness of fit test 

https://en.m.wikipedia.org/wiki/Goodness_of_fit

would tell you how good the model is.

If your samples are independent noise generation is straightforward .  A Bayesian model is different because you use another (prior) noise generator as an input to the fixed parameter noise generator.

If your samples are correlated you can use a Gibbs sampler. 

You should perform an independence test on your data, of which a Google search will provide a number of candidates 

You also might consider boot-strapping

https://en.wikipedia.org/wiki/Bootstrapping_(statistics)

It is also very likely that a physics based noise model exists for your device.