As suggested by <PERSON> this seems to be a normal-ordering issue. I will review here a toy example that was helpful for me. Consider a propagator in a free scalar field theory $$G^{m^2}(x,y)=\left<\phi(x)\phi(y)\right>=\frac{\int\mathcal{D}\phi e^{-S_{m^2}[\phi]}\phi(x)\phi(y)}{\int\mathcal{D}\phi e^{-S_{m^2}[\phi]}},\qquad S_{m^2}=\frac12\int \nabla\phi^2+m^2\phi^2$$ For any $m$ this propagator is <PERSON>'s function of the massive Laplacian $$(-\Delta+m^2)G^{m^2}(x,y)=\delta(x-y)$$ so we know what it is. Now let us replace $m^2\to m^2+\mu^2$ and treat $\mu^2$ as a small perturbation. Then we expect $$G^{m^2+\mu^2}(x,y)=G^{m^2}(x,y)+\mu^2\frac{\partial}{\partial m^2}G^{m^2}(x,y)+O(\mu^4)=\frac{\int\mathcal{D}\phi e^{-S_{m^2}[\phi]}\phi(x)\phi(y)\Big(1-\frac{\mu^2}2\int\phi^2+O(\mu^4)\Big)}{\int\mathcal{D}\phi e^{-S_{m^2}[\phi]}\Big(1-\frac{\mu^2}2\int\phi^2+O(\mu^4)\Big)}=\frac{\left<\phi(x)\phi(y)\right>-\frac{\mu^2}2\int_z\left<\phi(x)\phi(y)\phi^2(z)\right>+O(\mu^4)}{\left<1\right>-\frac{\mu^2}2\int_z\left<\phi^2(z)\right>+O(\mu^4)}$$ Although here the interaction term is just quadratic it still produces the problematic self-contractions. 818
This example also illustrates that maybe computing just the partition function (the denominator) is no good, it's better to consider some observable.
One way to deal with this formal problem is by assuming that fields coming from the action are normal-ordered. This excludes their self-contractions. In particular it implies that denominator just $\left<1\right>$ so that the first non-trivial correction is $$-\frac{\mu^2}2\int_z\left<\phi(x)\phi(y)\phi^2(z)\right>\to-\mu^2\int_z\left<\phi(x)\phi(z)\right>\left<\phi(y)\phi(z)\right>=-\mu^2 \int_z G(x,z)G(z,y)$$ which is finite. Interestingly, comparing with the direct expansion of $G^{m^2+\mu^2}$ this implies a relation for the propagator $$\partial_{m^2}G^{m^2}(x,y)=-\int_z G^{m^2}(x,z)G^{m^2}(z,y)$$ which can indeed be derived by differentiating the <PERSON> equation w.r.t. $m^2$.
So at least in this case the naive perturbation theory with the normal-ordering prescription seems to be a valid method. 818