Renormalization is only a condition for _perturbative_ well-definedness. Nonlinear transformations change the meaning of the perturbative expansion, hence may make a renormalizable theory nonrenormalizable and vice versa. In the (apart from its mathematically unresolved definition nonperturbative) path integral formulation, nonlinear transformations are very natural, generalizing nonlinear transformations of a finite-dimensional integral $\int_{R^n} e^{i S(x)} dx$ via the substitution rule.

Note that upon discretizing a field theory to a finite number of space-time points reduces the path integral to such an integral. In infinite dimensions (i.e., for a full QFT) subtle things must be taken account of to get valid results, and this is an art, not a science, as so far nobody has found a logically sound definition of the path integral for a nonquadratic action.