Alternative to using $R^2$ to assign data categories? A background to my problem: I use survey data on firms, where I want to measure the relationship between a binary variable (perceived growth barriers) and firm size. However, I cannot treat "firm size" as continuous, but I rather need to categorize firms. For this, I have chosen to categorize them based on their statistical relationship to the dependent variables.
My approach to fit categories has been to run regressions where I try out dummy variables for all consecutive firm size intervals using OLS (250 regressions per round). I have then categorized the first size category based on which of them has the highest $R^2$, after which I have repeated the process until all sizes are categorized.
However, my data exhibits high variance among larger firms, which means that I cannot use $R^2$ alone as it would only end up creating "overly wide" categories. Therefore, I have also weighted each $R^2$ output with the estimated Kernel density of the bandwidth where the categories end (e.g., a category containing firms with 4-16 employees would be weighted by the Kernel density of the size "16 employees"). This was made to "slow down" the regression algorithm and to force it to include influential groups that are relevant to my research.
However, this last solution was made ad-hoc and not with respect to previous research (on which I found none with respect to creating categories). 
My question is now:
Are there any alternative model fit measures to $R^2$ that is perhaps less sensitive to heteroscedasticity in the data? (i.e., ideally a measure that would not require the use of Kernel density weights to solve the above problem).
Alternatively, do you have any suggestions on improvements or alternative approaches to solving this issue? 
 A: You have a binary outcome ("experiences growth barrier" - yes/no), and a continuous predictor (firm size). You suspect a nonlinear relationship between the two.
Your best bet is a standard logistic regression. In order to model potential nonlinearities, do not feed firm size into the logistic regression as-is. Rather, transform them using splines. 
In a comment, you write:

marginal effects estimates cannot deliver anything else than a linear measure

This is incorrect - just use splines. These work for logistic regression just as well as for "vanilla" OLS. I have used splines to model nonlinearities in logistic regression models (regressing the likelihood to develop PTSD on spline transformed traumatic event load, Kolassa et al., 2010, J Clin Psych, and the same for the likelihood for spontaneous remission, Kolassa et al., 2010, Psych Trauma).
I very much recommend Frank Harrell's Regression Modeling Strategies on splines. 
Do not use discretization to model nonlinearities, since the discretization will introduce discontinuities that are typically spurious. (In your specific scenario, discontinuities may actually be valid for regulatory reasons; e.g., certain regulations on reporting or employee protection may only apply to firms of a certain size. If something like this is pertinent, add one or more Boolean indicator variables.)
A: An alternative approach I'm thinking of is to take the log of the firm size, instead of using it raw. That way, the difference between larger numbers gets smaller.
After that, my hunch is you will be able to simplify your classification algorithm, since your data will have been preprocessed to be more homoscedastic.
