We provide an "optimal fluctuation" approach which allows us to examine the statistics of stretched 2D fractal polymer chains near an impermeable disc. We find that the span of the polymer away from the surface scales with the Kardar-Parisi-Zhang (KPZ) growth exponent 1/3, for any fractal dimension of the polymer. We pay attention to the mathematical analogy of the model under consideration with 1D Balagurov-Vaks trapping problem related to 1D Anderson localization. In parallel we consider statistics of nonuniform 1D random walks  as a "mean-field" approximation of Edelman-Dimitriu approach to RMT and derive the KPZ scaling for the mean-square random walk displacement. We discuss possible applications of obtained results for transport properties in laminar flows of liquids in corrugated channels.