We compute the accuracy at which a Laser Interferometer Space Antenna-like space-based gravitational wave detector will be able to observe deviations from general relativity in the low frequency approximation. To do so, we introduce six correction parameters that account for modified gravity in the second post-Newtonian gravitational wave phase for inspiralling supermassive black hole binaries with spin precession on quasicircular orbits. Our implementation can be regarded as a subset of the parametrized post-Einsteinian formalism developed by Yunes and Pretorius, being able to investigate also next-to-leading order effects. In order to find error distributions for the alternative theory parameters, we use the Fisher information formalism and carry out Monte Carlo simulations for 17 different binary black hole mass configurations in the range 105Msun<M<108Msun with 103 randomly distributed points in the parameter space each, comparing the full (FWF) and restricted (RWF) versions of the gravitational waveform. We find that the binaries can roughly be separated into two groups: one with low (≲107Msun) and one with high total masses (≳107Msun). The RWF errors on the alternative theory parameters are 2 orders of magnitude higher than the FWF errors for high-mass binaries while almost comparable for low-mass binaries. Because of dilution of the available information, the accuracy of the binary parameters is reduced by factors of a few, except for the luminosity distance, which is affected more seriously in the high-mass regime. As an application as well as to compare our research with previous work, we compute an optimal lower bound on the graviton Compton wavelength, which is increased by a factor of ˜1.6 when using the FWF.