 Worldline variation before proper-time normalization

In the reparametrization-invariant massive-particle action, vary the square-root kinetic term before choosing proper time. Its velocity derivative gives $mc{u}_a/\sqrt{-u^2}$, which becomes $m{u}_a$ after imposing $u^2=-c^2$. Replacing the square root by a constant before variation would incorrectly remove the inertial momentum.