Consider you have some nonlinear function
\begin{align} y_i&=\epsilon_i f(\beta,x_i) \end{align}
where $\epsilon_i$ is log-normally distributed with mean 1, and \begin{equation} f(x,\beta)=\frac{\beta_1x}{1+\beta_2x} \end{equation}
One way to estimate $\beta_1$ and $\beta_2$ is to do nonlinear regression on the log transformed data using the following model
\begin{align} \log(y_i)&=\log(f(\beta,x_i)) + e_i \end{align}
where $e_i$ is now normally distributed (with a mean slightly below zero). However, given this saturating $f$ the log transformation squashes data close to the asymptote $\beta_1 / \beta_2$, which based on me playing around with simulated data can lead to poor fits, where the data closer to the origin seems to be counting too much towards the fit.
Based on this question there seems to be a trade-off between positively skewed data and the ill-effects of the transformation. My question is just how positively skewed does the error have to be for the benefits of taking log transform to outweigh the drawbacks if the error is log-normal? What other methods exist when the transformation is leading to fits you can tell are poor just by eyeballing the data? Is it best just to do a minimizing SSE calculation on the initial model simply ignoring that your data is positively skewed?