@fergusjchadwick: I just realized why the Pareto-K is almost guaranteed to be bad if you just add up the contributions to target from this parametrization. In this case, you don’t compute the log likelihood of the observed values given the linear predictors. You actually compute the log likelihood given the linear predictors AND the nuisance parameters. Since the observed values have huge influence on the associated nuisance parameters, loo correctly treats them as having large influence on the model and thus having high k-hat.
If you could integrate the nuisance parameters out, you would probably get a reasonably-behaved likelihood that would play nicely with loo. That could probably be done by implementing some Monte-Carlo integrator for multivariate normals in the generated quantities, but that could be hard to do (and expensive to compute)
It could also make some kind of weird sense to compute the multivariate normal log likelihood of the nuisance parameters given the linear predictors and feed that to loo, but I can’t think completely clearly if that would correspond to a meaningful quantity or not.