To do perturbation theory, one needs a fixed point (unperturbed theory). These are commonly free theories, for which the fixed points are called Gaussian points (because free theories are quadratic in fields, so that the path integral integrand resembles a Gaussian function). But this does not need to be the case necessarily. GR is not perturbative renormalizable around its Gaussian point, but it might be a fixed non-gaussian point around which one can carry out a renormalizable perturbation theory, as suggested by the Asymptotic Safety (AS) scenario. Whereas the existence of such a nontrivial fixed point is in principle possible, the counting of degrees of freedom connected with the BH entropy makes this scenario unlikely, as no local field theory in 4 dim such as AS can reproduce the BH entropy. The Gross-Neveu model is a typical example of a QFT with a non-trivial fixed point.
In addition, if one believes that QFT is just a framework to describe low-energy physics, existing a more fundamental framework such as strings, then one shouldn't necessarily be worried because the underlying framework should take care of the breakdown of the perturbative expansion in the low-energy QFT.