Due to its unique scalability potential, continuous-variable quantum optics is a promising platform for large-scale quantum computing. In particular, very large cluster states with a two-dimensional topology that are suitable for universal quantum computing and quantum simulation can be readily generated in a deterministic manner, and routes towards fault tolerance via bosonic quantum error correction are known. In this article we propose a complete measurement-based quantum computing architecture for the implementation of a universal set of gates on the recently generated two-dimensional cluster states [M. V. Larsen etal., Science 366, 369 (2019); W. Asavanant et al., Science 366, 373 (2019)]. We analyze the performance of the various quantum gates that are executed in these cluster states as well as in other two-dimensional cluster states (the bilayer-square lattice and quad-rail lattice cluster states [R. N. Alexander et al., Phys. Rev. A 94, 032327 (2016); N. C. Menicucci, Phys. Rev. A 83, 062314 (2011)]) by estimating and minimizing the associated stochastic noise addition as well as the resulting gate error probability. We compare the four different states and find that, although they all allow for universal computation, the quad-rail lattice cluster state performs better than the other three states, which all exhibit similar performance.