From the BO approximation, we have the product of the electronic [itex]\varphi_i(r,R)[/itex] and the nuclear [itex]\eta_i(R)[/itex] wavefunction. For the transition, we use Fermi's Golden rule, where the Dipole-Operator [itex]\mu[/itex] "initiates" the transition. So we end up in [itex]r_{i\rightarrow j}=\left\langle \eta_i(R) \varphi_i(r,R) |\mu| \varphi_j(r,R) \eta_j(R) \right\rangle[/itex]. Here we have an inner integral over the electron coordinates [itex]r[/itex] and an outer integral over the nuclei coordinates [itex]R[/itex]. It is important to note here that the inner integral [itex]\left\langle \varphi_i(r,R) |\mu| \varphi_j(r,R) \right\rangle[/itex] is a function of [itex]R[/itex]. The approximation is now that this inner integral is taken out of the outer interal, even though the former one is dependent of [itex]R[/itex] - which is the integration variable of the outer integral. Now the above equation looks like this: [itex]r_{i\rightarrow j}= \left\langle\varphi_i(r,R) |\mu| \varphi_j(r,R) \right\rangle \cdot\left\langle \eta_i(R)|\eta_j(R) \right\rangle[/itex] So actually the electronic integral is handled independently of the nuclei integral. The former one is a usual transition (with an operator for the transition according to Fermi's Golden rule), while the latter one is only an overlap of wavefunctions any more!
Schafspelz, the approximation you made can be justified further by using diabatic electronic states ##\eta_j## instead of the adiabatic electronic wavefunctions. The diabatic states depend only very little on R. A second step in the Franck-Condon approximation is to replace the dipole integral by a semiclassical expression so that only the neighbourhoods of the turning points of the nuclear motion contribute to the integrand.