This paper studied the weight function $w(x)=\pi^{\arctan(x)}$, the $\beta$ = 1 special case of the Romanovski family. We proved, as a structural theorem, that no closed-form Rodrigues representation or constant-eigenvalue Sturm–Liouville equation could hold for this weight on any compact interval case, because the unique Pearson polynomial $\sigma(x) = 1 + x^2$ has no real roots and thus no compact-interval endpoint-vanishing property. This placed the present case structurally outside the regime where the Romanovski construction was valid. We constructed the associated orthogonal polynomials (Keşan polynomials) directly from the three-term recurrence relation, which held independently of any differential representation, and verified all recurrence coefficients and cross-orthogonality to machine precision. We then applied this machinery to the Gompertz–Burgers tumor invasion model. The principal computational finding, independently and reproducibly established, was that single-domain polynomial collocation failed entirely for this problem at physiological diffusion parameters ($\nu = 10^{-3}$), due to a transition layer of width approximately 0.2% of the domain. Two-domain Chebyshev decomposition reduced the weighted mean-squared error (WMSE) by a factor of 19–21, relative to the best achievable single-domain result. The basis choice (Keşan vs. Legendre vs. Chebyshev) within the decomposition was immaterial; the critical variable was the decomposition strategy and interface placement, not the polynomial family.