Nonlinear response signatures are increasingly recognized as useful probes of condensed matter systems, in particular for the characterization of topologically nontrivial states. The circular photogalvanic effect (CPGE) is particularly useful in the study of topological semimetals, as the CPGE tensor quantizes for well-isolated topological degeneracies in strictly linearly dispersing band structures. Here, we study multiplicative Weyl semimetal band structures, and find that the multiplicative structure robustly protects the quantization of the CPGE even in the case of nonlinear dispersion. Computing phase diagrams as a function of Weyl node tilting, we find a variety of quantized values for the CPGE tensor, revealing that the CPGE is also a useful tool in detecting and characterizing the parent topology of multiplicative topological states.