How would we know that Dirac equation does not describe composite spin-1/2 fermions?

The Dirac equation does describe composite spin-1/2 fermions - namely, baryons like the proton and the neutron. Conversely, future experiments might reveal the electron to be composite even though it's described by the Dirac equation (plus perturbative corrections).

The vertex term you describe does appear in the scattering cross-section for proton-photon scattering, but it's corrected by loop-level renormalization terms that stem from interactions, which are tiny (but measurable) for the electron but large for the proton.

Just as a complement to tparkers answer, people have been using the Dirac equation for composite particles since very long ago. Just remind yourself of the Yukawa model for hadron-hadron interactions

$$\mathcal L=\bar\psi(i\gamma^\mu\partial_\mu-M)\psi+\frac{1}{2}(\partial^\mu\phi)(\partial_\mu\phi)-\frac{1}{2}m^2\phi^2-ig\phi\bar\psi\gamma^5\psi$$

Note the first term, which is the Dirac equation for the nucleon in question. This theory is what gives us the attractive Yukawa potential

$$V(r)=-\frac{g^2}{4\pi}\frac{e^{-mr}}{r}$$

One succes of the Dirac equation is that it correctly implies the particle g-factor to be g=2, explaining the lepton g-factor. For protons and neutrons g is very different from 2, so the Dirac equation itself cannot be applied to these.

The squared Dirac equation exhibits a spin dependent term, the relativistic generalization of the Pauli interaction. In this equation the g-factor of 2 can be substituted for by the g-factor of the proton or the neutron. In this case the modification takes account of the fact that these are composite particles. There are also loop corrections to the g-factor. These can also be taken into account in this way - but avoid double counting in perturbation theory. So the answer is: both.