  Adjusting a previously developed Grotch-type approach to a perturbative
calculation of the electronic vacuum-polarization effects in muonic atoms, we
find here the two-loop vacuum polarization relativistic recoil correction of
order $\alpha^2(Z\alpha)^4m^2/M$ in light muonic atoms. The result is in
perfect agreement with the one previously obtained within the Breit-type
approach. We also discuss here simple approximations of the irreducible part of
the two-loop vacuum-polarization dispersion density, which was applied to test
our calculations and can be useful for other evaluations with an uncertainty
better than 1%.
