Flexible random scale-mixture models provide a framework for capturing a broad range of extremal dependence structures. However, likelihood-based inference under the peaks-over-threshold setting is often computationally infeasible, due to the censored likelihood requiring repeated evaluation of high-dimensional Gaussian distribution functions. We propose a multiplicative log-Laplace nugget that yields conditional independence in the censored likelihood, resulting in a joint likelihood function that is the product of univariate densities which are available in closed form. This eliminates multivariate Gaussian distribution function evaluations and thereby enables inference for threshold exceedances in high dimensions, which represents a major shift for spatial extremes modelling as the total computational cost is now primarily driven by standard spatial statistics operations. We further show that a broad class of scale-mixture processes augmented with the proposed nugget preserves the extremal dependence structure of the underlying smooth process. The proposed methodology is illustrated through simulation studies and an application to precipitation extremes.

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