We study a classical nonlinear Duffing oscillator driven by Gaussian white noise by developing a self-consistent Born approximation (SCBA) within a field-theoretic framework. In analogy with particle field theory, we construct a self-consistent modal mean-field solution that renormalizes the oscillator's natural frequency and reproduces the characteristic amplitude-frequency dependence of such systems. At the Hartree level, this mean field captures all static interactions among the noise-activated internal Fourier modes (NAIFMs). By subsequently incorporating the Born approximation, we naturally include dynamic correlations between NAIFMs, which substantially improve the description in the large-amplitude regime where nonlinear effects become prominent. We show that standard perturbative expansions-particularly those at one and two loops-fail to describe observables such as the mean-square displacement (MSD) in this regime, exhibiting a breakdown of the expansion. In contrast, the SCBA accurately reproduces both the MSD and the renormalized frequency over a broad amplitude range, in excellent agreement with numerical simulations. This approach provides a robust analytical framework for nonlinear oscillators under stochastic driving, with direct relevance to micro- and nanomechanical resonators.