I'm struggling to find a goodness of fit test of the above. The non-parametric tests I have looked at (KS) seem to be unable to deal with estimated parameters - can someone help?
Unfortunately, there is no simple answer to your problem. The KS test methodology is generally used, but the critical regions need to be determined via simulation if you are not trying to test the normality of the data. See this link for details, it explains how one does this for non-normal tests. This paper is also very helpful, as it addresses exactly how you calculate these bounds using a computer.
For dealing with non-normality, you could consider using the Anderson-Darling test if you have an idea what your hypothesized distribution function is. This test is implemented in R in the ADGofTest. I know for testing normality there are modifications made to the test statistic to account for estimated parameters. I haven't thoroughly researched other distributions, but I would assume it is possible to make similar adjustments. The other option would be to just bootstrap the test statistic to estimate its distribution.
After some cursory research, it also looks like there is a non-parametric version of this test. See this paper: http://www.cithep.caltech.edu/~fcp/statistics/hypothesisTest/PoissonConsistency/ScholzStephens1987.pdf
It appears this test is also be implemented in R, in the package kSamples: http://cran.r-project.org/web/packages/kSamples/kSamples.pdf