Many scientific inquiries in natural sciences involve approximating a spherical field –namely a scalar quantity defined over a continuum of directions– from generalised samples of the latter. Typically, a convex optimisation problem is formulated in terms of a data-fidelity and regularisation trade-off. To solve this optimisation problem numerically, scientists resort to discretisation via spherical pixelisation schemes called tessellations. Finite-difference methods for approximating (pseudo-)differential operators on spherical tessellations are however unavailable in general, making it hard to work with generalised Tikhonov or Total Variation (gTV) regularisers. To overcome such limitations, canonical spline-based discretisation schemes have been proposed. In the case of Tikhonov regularisation, optimality has been proven for spherical interpolation. A similar result for gTV regularisation is however still lacking. In this work, we propose a spline approximation framework for a generic class of reconstruction problems on the hypersphere. Such problems are formulated over infinite-dimensional Banach spaces, seeking spherical fields with minimal gTV and verifying a convex data-fidelity constraint. The data itself can be acquired by generalised sampling strategies. Via a novel representer theorem, we characterise their solution sets in terms of spherical splines with sparse innovations, Green functions of the gTV pseudo-differential operator. We use this result to derive an approximate canonical spline-based discretisation scheme, with controlled approximation error. To solve the resulting finite-dimensional optimisation problem, we propose an efficient primal-dual splitting method. We illustrate the versatility of our framework on numerous real-life examples from the field of environmental sciences and radio astronomy.