In Structural Health Monitoring (SHM), the recovery of distributed mechanical parameters from sparse data is often ill-posed, raising critical questions about identifiability and the reliability of inferred states. While deterministic regularization methods such as Tikhonov stabilize the inversion, they provide little insight into the spatial limits of resolution or the inherent uncertainty of the solution. This paper presents a Bayesian inverse framework that rigorously quantifies these limits, using the identification of distributed flexural rigidity from rotation (tilt) influence lines as a primary case study. Fisher information is employed as a diagnostic metric to quantify sensor informativeness, revealing how specific sensor layouts and load paths constrain the recoverable spatial features of the parameter field.The methodology is applied to the full-scale openLAB research bridge (TU Dresden) using data from controlled vehicle passages. Beyond estimating the flexural rigidity profile, the Bayesian formulation produces credible intervals that expose regions of practical non-identifiability, which deterministic methods may obscure. The results demonstrate that while the measurement data carry high information content for the target parameters, their utility is spatially heterogeneous and strictly bounded by the experiment design. The proposed framework unifies identification with uncertainty quantification, providing a rigorous basis for optimizing sensor placement and interpreting the credibility of SHM diagnostics.