Replying as the author -- I do spend some time discussing hetereoskedastic noise (beginning in §2.2 and intermittedly throughout following chapters), although you're right that I don't discuss this particular modeling approach. Personally I think that inferring hetereoskedastic noise from data alone during Bayesian optimization is likely to be very difficult, as you'll need either a lot of data and/or to be in a very small dimension in order to identify the variable noise scale. (Note that the example in the hetGP writeup is only in one dimension.)
However, when the noise scale is either variable (but known) or can be modeled with a relatively simple (e.g., parametric) model, there may be some benefit to the added model complexity. Here you could include the parameters of the noise model into the model hyperparameters and proceed following the discussion in chapter 4. In doing so, I would be careful to ensure that the data actually support the heteroskedastic noise hypothesis.
Another approach that might be useful in some contexts is a heavy-tailed noise model such as Student-t errors (§§ 2.8, 11.9, 11.11).
However, when the noise scale is either variable (but known) or can be modeled with a relatively simple (e.g., parametric) model, there may be some benefit to the added model complexity. Here you could include the parameters of the noise model into the model hyperparameters and proceed following the discussion in chapter 4. In doing so, I would be careful to ensure that the data actually support the heteroskedastic noise hypothesis.
Another approach that might be useful in some contexts is a heavy-tailed noise model such as Student-t errors (§§ 2.8, 11.9, 11.11).