A site-specific Type-II codebook design is proposed for downlink massive multiple-input multiple-output (MIMO) limited-feedback beamforming. The key idea is to embed a learned site-specific propagation prior into the Type-II channel state information (CSI) feedback pipeline. Specifically, the base station (BS) uses a low-overhead reference signal received power (RSRP) fingerprint collected during synchronization signal block (SSB) probing to infer a user equipment (UE)-dependent dominant beam subspace before explicit CSI acquisition. The UE then estimates and feeds back only the low-dimensional effective channel coefficients within this inferred subspace, thereby avoiding full-dimensional online subspace discovery while retaining a rich multi-beam representation capability. To analyze the proposed design and compare it with standardized feedback mechanisms, a unified subspace-projection framework is developed by jointly characterizing CSI acquisition, UE-side compression, BS-side reconstruction, and effective spectral efficiency. Under this framework, Type-I, Type-II, port-selection feedback, and the proposed scheme are interpreted as different ways of inducing a feedback representation subspace. The probing codebook and the BS-side subspace inference network are then formulated as a coupled task-oriented design problem and are optimized end-to-end by maximizing the normalized CSI-capture efficiency. Extensive simulation results demonstrate that the proposed feedback scheme achieves Type-II-comparable CSI-capture capability with substantially lower online overhead and UE-side complexity, thereby improving the effective spectral efficiency.