We investigate the Mott-superfluid transition in a two-component Bose-Hubbard model with local Rabi coupling, focusing on the asymmetric regime where interaction, hopping, and chemical-potential imbalances compete with coherent interconversion. We show that the transition is governed by the softening of the lowest eigenmode of the full two-component response kernel rather than by the atomic gaps of either component separately. This eigenmode-based description clarifies the role of the symmetric common-mode channel and shows how local Rabi mixing reorganizes the Mott boundaries beyond the conventional two-component Bose-Hubbard scenario. Combining an analytical common-mode reduction with exact local susceptibility calculations and numerical determination of the phase boundary, we demonstrate that the first-lobe tip can evolve nonmonotonically with increasing Rabi coupling. Its minimum coincides with the maximum of the dominant susceptibility mode, signaling a crossover from softening to restiffening of the critical collective response. These results provide a susceptibility-based interpretation of Mott criticality in Rabi-coupled lattice bosons and identify clear signatures distinguishing the Rabi-coupled system from the conventional two-component Bose-Hubbard model.

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• Bose gases
• Bose-Bose mixtures
• Cold atoms & matter waves
• Cold gases in optical lattices
• Collective effects in atomic physics
• Critical phenomena
• Decoherence in quantum gases
• Interatomic & molecular potentials
• Josephson effect
• Path integrals
• Quantum statistical mechanics
• Spin-orbit coupling
• Superfluidity
• Rabi model