Is Higgs mass relativistic? Massive elementary particles receive their mass via interaction with the Higgs field. This answer offers some explanation.
But this only accounts for one percent or so of the mass in the Universe. The rest is supplied by the binding energy which pulls the particles together into nucleons and atoms, according to $E = mc^2$.
So mass appears to have a dual nature; part Higgs mass and part relativistic mass. Is this correct, and if so then does Higgs mass contribute to the curvature of spacetime? Or does Higgs mass also derive from $E = mc^2$ via the energy of the Higgs field (in which case it must contribute to the curvature of spacetime)?
 A: All mass is relativistic in nature, to the extent it is a parameter in all relativistic behavior of everything, including the limit of slowly moving and stationary objects and particles.
The Higgs QFT component is relativistically invariant, like everything else in QFT.
The Higgs quantum field interactions  underlie all the mass we know of to the W and Z gauge bosons, and all leptons, like the electron, the μ and τ, and possibly the neutrinos. They also give small masses to all quarks; but the bulk of the mass of all hadrons comes from an elaborate mechanism in QCD, interactions of gluons in a completely independent interaction.
Not just all masses, but all types of energy, as well, couple to gravity, and, as such, contribute to spacetime curvature.
When it comes to cosmic scales, most of the mass of the universe is in mysterious "dark energy" and "dark matter" sectors, (arguably) likely independent of Higgs interactions.
Frankly, I had trouble parsing out the logical trail-map of implications you may well be asking about.
A: A search-term-only answer: if Higgs-field masses didn’t contribute to the curvature of spacetime, it would be a violation of the relativistic “equivalence principle.”
All searches for equivalence-principle violations have been either inconclusive or negative: there is no solid evidence against this bedrock tenet of relativity. But devising new tests is hard. The embarrassingly large uncertainty on the gravitational constant $G$ could (in principle) hide a lot of new physics.
A: You are asking "The rest is supplied by the binding energy which pulls the particles together into nucleons and atoms, according to E=mc2. So mass appears to have a dual nature; part Higgs mass and part relativistic mass. Is this correct, and if so then does Higgs mass contribute to the curvature of spacetime?". The answer is yes, in your example, the proton's and neutron's rest mass is 99% binding energy, and only 1% the comes from the rest mass of the constituent quarks, but that 1% still contributes to the stress-energy tensor and creates spacetime curvature.

If you use General Relativity instead you'll find that photons make a contribution to the stress energy tensor, and therefore to the curvature of space.

Does a photon exert a gravitational pull?
Please note that even a single quark, or electron (getting its rest mass from the Higgs mechanism as you say) does create spacetime curvature, and even a photon (being massless) does bend spacetime. Ultimately it is because all of them possess stress-energy.
