The recently proposed chameleonic extension of bigravity theory, by including a scalar field dependence in the graviton potential, avoids several fine-tunings found to be necessary in usual massive bigravity. In particular it ensures that the Higuchi bound is satisfied at all scales, that no Vainshtein mechanism is needed to satisfy Solar System experiments, and that the strong coupling scale is always above the scale of cosmological interest all the way up to the early Universe. This paper extends the previous work by presenting a stable example of cosmology in the chameleon bigravity model. We find a set of initial conditions and parameters such that the derived stability conditions on general flat Friedmann background are satisfied at all times. The evolution goes through radiation-dominated, matter-dominated, and de Sitter eras. We argue that the parameter space allowing for such a stable evolution may be large enough to encompass an observationally viable evolution. We also argue that our model satisfies all known constraints due to gravitational wave observations so far and thus can be considered as a unique testing ground of gravitational wave phenomenologies in bimetric theories of gravity.