Yeah it's a great question. I don't know the answer, but I suspect the people who study it strongly suspect that it is highly complex in this sense. Otherwise they would be looking for simpler representations instead of running massive simulations.

To your question, I think there is an elegant answer actually; most composite particles in QCD are unstable. They're either made out of equal parts matter and antimatter (like pions) or they're heavier than the proton, in which case they can decay into one (or more) protons (or antiprotons). If any of the internal complexities of the proton made it distinguishable from other protons, they wouldn't both be protons, and one could decay into the other. Quantum mechanics also helps to keep things simple by forcing the various properties of bound states to be quantized; there isn't a version of a proton where e.g. one of the quarks has a little more energy, similar to how the energies atomic orbitals are quantized.