Hyperinvariant tensor networks (hyMERA) were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence. Although this new class of network shows much potential for simulating conformal field theories on hyperbolic bulk manifolds with quasiperiodic boundaries, many issues must still be solved. In this manuscript we analyze the challenges related to optimizing tensors in a hyMERA with respect to some critical spin chain, and compare with standard approaches in MERA. Additionally, we show two new sets of tensor decompositions which exhibit different properties from the original construction, and imply that the multitensor constraints are more general than previously suggested. Lastly, we provide numerical evidence that the constraints imposed on the spectra of local descending superoperators in hyMERA are compatible with the operator spectra for many minimial model CFTs.