The Lamperti transform offers a powerful bridge between self-similar processes and stationary dynamics, making it especially useful for analyzing anomalous diffusion models that lack stationary increments. In this paper we examine the Lamperti transforms of scaled sub-fractional and bi-fractional Brownian motions, deriving explicit covariance formulas, asymptotic behaviour, and precise exponential mixing rates. We also introduce Langevin type integral processes driven by these Gaussian fields, identify their self-similarity exponents, and show that their Lamperti images again form stationary Gaussian processes with rapid decorrelation. Through inverse Lamperti relations and Birkhoff's theorem, we establish rigorous single trajectory reconstruction of ensemble quantities for the original non-stationary processes. The results extend the scope of the scaled Lamperti framework to Gaussian processes with non-stationary increments and richer dependence structures.