Majorana zero modes in networks of one-dimensional topological superconductors obey non-Abelian braiding statistics. Braiding manipulations can be realized by controlling Coulomb couplings in hybrid Majorana-transmon devices. However, strong disorder may induce accidental Majorana modes, which are expected to have detrimental effects on braiding statistics. Nevertheless, we show that the Coulomb-assisted braiding protocol is efficiently realized also in the presence of accidental modes. The errors occurring during the braiding cycle are small if the couplings of the computational Majorana modes to the accidental ones are much weaker than the maximum Coulomb coupling.