We investigate the weakly nonlinear dynamics of the flutter instability of a planar Cosserat rod in a viscous fluid driven by a terminal follower force. Building on our previous linear stability analysis, we derive Stuart-Landau amplitude equations governing the emergence and saturation of oscillations near the Hopf bifurcation. The Landau coefficients are expressed in terms of the critical eigenmode, its adjoint, and quadratic correction fields arising from the asymptotic expansion. The reduced theory predicts the criticality of the Hopf bifurcations, the saturated oscillation amplitude, and the nonlinear frequency correction, in quantitative agreement with direct numerical simulations for representative material parameter regimes. Applying the theory to both stability boundaries of the finite oscillatory window, we find that the lower Hopf bifurcations are supercritical, whereas the upper restabilization threshold is subcritical. These results provide a normal-form description of follower-force-driven oscillations in geometrically exact Cosserat rods and a reduced framework for studying nonconservative elastohydrodynamic instabilities in soft robotic structures and active filaments.