Thanks for the suggestion @DanhPhan! Actually, I was originally thinking I might need the LatentKron GP since the covariance structure could be considered separable along the sampling (i.e., non-spatial) dimension. But this would result in a 3d GP, and I think the parameter prior should really only incorporate variation in 2 dimensions (with repeats).
Thanks for sharing the blog post as well! Really cool to see them build compatibility for a spherical domain.