We investigate the field-dependent low-energy thermodynamics of the distorted triangular quantum antiferromagnet Cu₃(OH)₄(HCO₂)₂ using a sector-resolved Superblock Diagonalization Method (SBDM) supplemented by a transfer-matrix treatment of weakly coupled layers. The strong exchange hierarchy, dominated by the Cu1–Cu1 intradimer coupling J₂=150 K, separates a high-energy dimer sector from a much softer magnetic manifold formed predominantly by the Cu2 moments. In the 24-site cluster, sixteen Cu1 spins form the strongly bound sector while eight Cu2 spins remain magnetically active; polarization of these eight spins gives Sᶻ=4 compared with Sᶻ_(rm sat)=12, providing a direct microscopic origin for the one-third magnetization scale. The finite-temperature thermodynamics reveals a non-monotonic field evolution of the low-energy scale: the dominant C/T feature softens with increasing field, reaches a minimum near the field region around 2 T, and subsequently hardens as the low-temperature magnetization approaches M_(rm sat)/3. Temperature and field sweeps thus expose a common field-induced spectral reconstruction, while the strongly reduced entropy reflects the restricted number of thermally active degrees of freedom below the dimer excitation scale. Our results identify strong-dimer-induced reduction of the active magnetic Hilbert space, followed by field-driven reorganization of the residual spin sector, as the common microscopic origin of the one-third magnetic response and the non-monotonic low-temperature thermodynamics.