We investigate the single-particle momentum distribution, the two-body contact and three-body tail coefficients of mass-imbalanced three-body systems at the critical dimension $D_c$, where the transition between discrete and continuous scale invariance takes place as the spatial dimension is tuned between three and two dimensions. We show that the asymptotic momentum distribution at $D_c$ is governed by a distinct logarithmic scaling structure, which differs fundamentally from both the log-periodic behavior of Efimov states and the power-law scaling of the unatomic regime. This structure contains an additional quadratic-logarithmic contribution, characterized by the asymptotic coefficient $C_3’’$ , which leads to a finite and well-defined description of the momentum tail at the transition. The coefficient $C_3’’$ depends sensitively on the mass imbalance, changing sign across different mass configurations and vanishing for identical particles. For heavy-light-light configurations, the subleading three-body term vanishes at a characteristic momentum, leaving the leading two-body contribution at the order retained in the asymptotic expansion. The physical accessibility of this cancellation depends on the separation between the trimer size and the interaction range. We further analyze the narrow intermediate region connecting the Efimov and unatomic regimes, here identified as an intermediate scaling regime, whose extent and properties are strongly controlled by the mass ratio. These results establish the critical dimension as a regime with emergent scaling properties and provide quantitative momentum-space signatures for probing the transition between discrete and continuous scale invariance in few-body quantum systems.