I think the problem here is the gross oversimplification that "Higgs causes mass". The rest energy of a system is the mass of that system.
Consider the proton. Yes, some mass comes from interaction with the Higgs, but the vast majority comes from interactions mediated by gluons.
Gravity couples to the stress energy tensor. For the equivalence principle to hold, it can't "care" what the different contributions to the stress energy tensor are. And the Higgs is not the sole contributor let alone a major contributor, so it is not the cause of gravity, nor really even a real big cause of the 'source' of gravity.
For those QFT savy, would it be correct to say that with a non-zero vaccuum expectation of the Higgs, it is a redefinition of the fields (of massless fermions) to include these interactions which yields the "massive" fermion fields? ie. that the fermions are still massless per se, it is just mathematically convenient to subsume this interaction into the definition of the field? Am I even close?