I would like to know which (non-linear) transformation of my DV yields a better model (not necessarily with the aim of producing a DV that better fits the assumptions of the test). My impression is that the Log Likelihood is dependent on the scale of the DV such that I can not compare models based on the Log Likelihood (or AIC/BIC for that matter). I suspect that I could do an inverse transformation on each of the IVs relative to the transformation I was considering on the DV, but a) I'm not entirely sure and b) it seems like there must be a better way. Would the proposed method work? Is there a better way?
What do you mean by a "better" model? How to measure a model depends both on the type of model and what you want to improve, but, e.g. for a OLS regression you could use $R^2$. There are pseudo $R^2$ measures for some other models. You could look at the differences (measured in various ways) between the predicted and actual values. For logistic regression there are sensitivity and specificity. etc.
Fishing around for a transformation that improves the model (however that is measured) is kind of fishy. You should transform your data based on a) Interpretative sense and b) Model assumptions. What you are proposing is likely to be too dependent on the particular set of data, and also likely to be a model that is hard to interpret (e.g. the .314 power of a variable is not going to mean much).
If you really want to do this, you should at least separate your data into training and test sets, and report fit only for the test set.