First of all more than 99.9% of the mass around you is unrelated to the Higgs. It comes from the nuclear binding energy within neutrons and protons which the Higg's doesn't really contribute to. In a typical atom the Higgs is only giving the electrons mass, which are less than 0.06% of an atom's mass.

Secondly, quantum mechanics and relativity function perfectly fine together, that's what quantum field theory is about. Regardless of whether field one is considering the Higgs or not quantum field theory takes care of combining the principles of quantum mechanics and relativity, both special and general. So there is no problem with the quantum particles having relativistic mass or causing spacetime curvature. For a long time we've had the ability to write down theories where quantum matter causes gravity/spacetime curvature. It doesn't really matter if that mass comes from the Higgs or nuclear binding energy or whatever.

The issue with quantum gravity is a seperate one and has two parts.

(i) How to make gravity/spacetime curvature itself quantum. Even this is not a major issue, there do exist quantum treatments of gravity. It's more a problem of getting experimental evidence to compare them with. So for example there is no real issue with treating the Sun's gravitational field quantum mechanically, but the differences this makes from classical gravity are minute.

(ii) How quantum theory is affected by event horizons, either those of black holes or cosmological ones. Horizons restrict ones ability to make measurements, e.g. somebody outside a black hole cannot measure what is inside a black hole. Since quantum mechanics is all about what one can measure and what probabilities those measurements have, we know this affects QM in some way, but we don't know exactly how.

Probably in the full theory questions like "What is the probability there are X quasars in the Virgo supercluster?" will not be well-defined since no observer could actual measure this experimentally due to cosmological expansion and the theory will instead return "undefined" as the answer. This is similar to how in quantum electrodynamics the theory returns "undefined" for the question "what is the probability an electron will be in this region" and only returns a well defined answer for "what is the probability a detector in this region will activate".

Strictly speaking this would mean even natural questions one might ruminate on like "What is the chance there is at least one habitable world per galaxy in the Virgo Supercluster?" would be banned/nonsensical according to theory.