Rapid adiabatic passage, which enables complete population transfer via a frequency-chirped laser pulse, is valued for its robustness to pulse-area fluctuations in quantum-dot control. Previous work showed that strong coupling to a super-Ohmic phonon bath can further stabilize this robustness in a two-level system, an effect invisible to Markovian treatments. We extend this framework to a three-level quantum-dot-like ladder (ground state, single exciton, biexciton) coupled to a super-Ohmic phonon bath, using the multiple Davydov D2 Ansatz with the Dirac-Frenkel variational principle. Systematic multi-dimensional sweeps over pulse area, temporal chirp, and bath coupling strength reveal a qualitative bifurcation governed by the ladder’’s resonance structure. For a detuned ladder, the bath stabilizes and smooths population transfer into the single-exciton state, sustaining near-unity fidelity across all coupling strengths studied and preferring positive chirp. This extends the two-level bath-stabilisation picture to three-level systems. In contrast, for a double-resonant ladder, the same bath coupling disrupts coherent transfer: population is distributed across all three levels, the maximum single-exciton fidelity becomes non-monotonic in coupling strength, and the preferred chirp direction reverses relative to the detuned case. The non-monotonic dependence arises from bath-mediated incoherent redistribution from the biexciton back to the single-exciton state-a mechanism absent in two-level models and inaccessible to perturbative treatments. The observed chirp-sign reversal constitutes a falsifiable experimental prediction for distinguishing the two regimes without requiring independent spectroscopic knowledge of the biexciton binding energy.