  We calculate the two-photon exchange correction to the Lamb shift in muonic
helium-3 ions within the dispersion relations framework. Part of the effort
entailed making analytic fits to the electron-$^3$He quasielastic scattering
data set, for purposes of doing the dispersion integrals. Our result is that
the energy of the 2$S$ state is shifted downwards by two-photon exchange
effects by 15.14(49) meV, in good accord with the result obtained from a
potential model and effective field theory calculation.
