Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, even in moderate dimensions, due to the necessity of repeatedly evaluating the multivariate Gaussian distribution function. In this work, we attempt to achieve truly high-dimensional inference for extremes of spatial processes, while retaining the desirable flexibility in the tail dependence structure, by modifying an established class of models based on scale mixtures Gaussian processes. We show that the desired extremal dependence properties from the original models are preserved under the modification, and demonstrate that the corresponding Bayesian hierarchical model does not involve the expensive computation of the multivariate Gaussian distribution function. We fit our model to exceedances of a high threshold, and perform coverage analyses and cross-model checks to validate its ability to capture different types of tail characteristics. We use a standard adaptive Metropolis algorithm for model fitting, and further accelerate the computation via parallelization and Rcpp. Lastly, we apply the model to a dataset of a fire threat index on the Great Plains region of the US, which is vulnerable to massively destructive wildfires. We find that the joint tail of the fire threat index exhibits a decaying dependence structure that cannot be captured by limiting extreme value models.