Adjusting AIC for a model with a transformed response

I would like to compare two models, one of which has a transformation on the response variable, that is:

First model: $y \sim x+z$

Second model: $y^{1/k} \sim x+z$

Since AIC can be used only for models with the same form on the response, how could I potentially solve this problem? I know that if we have a log transformation on normal, we can correct it by adding twice the sum of log response, but how would we deal with a generalised case when y is transformed by raising it to the power of $1/k$, where $k\in\mathbb{N_+}$?

I also suspect you should take the uncertainty around the regression coefficients by treating estimate $\pm$ SE as a quasi-posterior - i.e. for each predicted dataset draw the coefficients from N(estimate, SE$^2$). That’s again, because the error goes into a non-linear transformation.