In this work, we propose a unified theoretical and practical spherical approximation framework for functional inverse problems on the hypersphere. More specifically, we consider recovering spherical fields directly in the continuous domain using functional penalised basis pursuit problems with gTV regularisation terms. Our framework is compatible with various measurement types as well as non-differentiable convex cost functionals. Via a novel representer theorem, we characterise their solution sets in terms of spherical splines with sparse innovations. We use this result to derive an approximate canonical spline-based discretisation scheme, with vanishing approximation error. To solve the resulting finite-dimensional optimisation problem, we propose an efficient and provably convergent primal-dual splitting algorithm. We illustrate the versatility of our framework on real-life examples from the field of environmental sciences.