Step (2) is incorrect, given the results of step(1). Given step (1), "theory" says either:
(a) Chose an appropriate likelihood function, based on an acceptable distribution. Use a large sample argument to justify a chi-squared test based on a likelihood ratio test.
(b) Choose an appropriate objective function (goodness-of-fit measure), such as a mean absolute difference. (Although this might need to be modified if you are fitting both location and scale.) Construct a test statistic based on this objective function, such as the improvement in the objective function on moving to the wider model. Construct critical values for the test statistic by undertaking a simulation study based on what you think are acceptable null distributions.
If you were happy enough to do a Bayesian analysis, you might note that several recent works have been implemented with structures where the "normal distribution" assumption has been replaced by a Student's t distribution with fixed but low degrees of freedom, which includes the Cauchy distribution. Hence there is a good chance that you could find a Bayesian analysis package that includes this facility, and this might prove a viable route for you. Of course, you might find a"frequentist" package to do something similar, if you need to look for pre-existing code ... you might look under "general linear model".