Is it possible to have a theory that treats both GR and QFT (e.g. QFT on a curved spacetime dynamically influenced by the standard QFT fields)? Is such a theory at least self-consistent (even if it does not apply to nature)? Or is there some fundamental incompatibility we run into without even trying to quantize GR (or perhaps we are somehow forced to quantize GR for consistency)?
This formalism exists, and is called semiclassical gravity. You can calculate effects using it, most notably, the existence of Hawking radiation. As an exact solution, it is somewhat unsatisfactory for these reasons:
First, it is inherently not an exact solution to anything. You treat the quantum field as if the background metric was an exact solution to GR. This invariably causes the field to bend and behave differently. This, obviously, will affect the background metric, which requires that you recalculate the metric, which then requires that you recalculate the field, etc. It's unclear whether this series of successive approximations even will converge to a stable solution. In practical cases, people usually only consider a small number of orders of this backreaction.
There are technical issues. In particular, QFT in a curved spacetime is only well-defined for a special class of background metric. In particular, QFT formalism is heavily dependent on "in" and "out" states. If we're going to rely upon intuition from QFT in Minkowski space, we need a region where the spacetime is flat to compute these states.
Even if we were to relax the previous requirement, we still need a timelike killing vector at infinity in order to define positive and negative frequency states, which is necessary if we are going to normal order our fock space (in non-technical language, consistently define the energy of the vacuum and what states are "particles" and which are "antiparticles")
Finally, there is just an inherent technical issue here, where we can pretty easily violate causality by combining QM effects with GR. For instance, imagine an experiment where a single (very massive) particle is sent through a slit, and after some time, will have a wavefunction whose location is two disjoint regions of space. Is the field a superposition of a mass distribution in both locations? What happens after the wavefucntion is collapsed? Does the metric change non-continuously? How do you resolve the singularity?
You could imagine something similar using the EPR experiment, or the like. The main problem is that QM admits non-local effects (albiet non-causal ones), while GR does not. How do you resolve this while naïvely coupling them?