This paper develops a unified operational calculus for composite Hankel–Laplace integral operators defined over a class of non-separable, weighted Sobolev-type testing function spaces . We establish the structural topologies of these test spaces through a countable collection of weight-restricted seminorms, proving completeness, metrizability, nuclearity, and Montel properties to confirm their Fréchet space classification. The topological dual space  is fully characterized, providing explicit polynomial-exponential growth conditions for continuous linear functionals. Kernel representation theorems and convolution algebra structures are formulated within the distributional setting. To demonstrate operational efficiency, we apply the generalized transform to solve dynamic fractional initial-boundary value problems governing anomalous radial transport and acoustic wave propagation in heterogeneous media. Exact distributional solutions are constructed via multivariable Fox -functions and three-parameter Wright-type hyper-Bessel functions. Furthermore, a stable numerical inversion scheme utilizing adaptive Gaver–Stehfest and parabolic contour techniques is formulated.