Luca Martino pointed out to me my own and forgotten review of a 2012 paper of his, “On the Generalized Ratio of Uniforms as a Combination of Transformed Rejection and Extended Inverse of Density Sampling” that obviously discusses a generalised version of Kinderman and Monahan’s (1977) ratio-of-uniform method. And further points out the earlier 1991 paper by Jon Wakefield, Alan Gelfand and Adrian Smith that contains the general form I rediscovered a few posts ago… Called the GRoU in Martino et al.. While the generalisation in the massive arXiv document is in finding Φ such that the above region is bounded and can be explored by uniform sampling over a box.
Neither reference mentions using the cdf transform, though, which does guarantee a bounded ratio-of-uniform set in u. (An apparent contradiction with Martino et al. statement (34), unless I am confused. Maybe due to using Φ⁻¹ instead of Φ?) But I still wonder at the usefulness of my derivations those past weeks!