It doesn't make a difference to me if it is a normaliser or a field formatter, I just need my json export to be able to render paragraphs as arrays properly. At first I thought the field formatter approach would be best, but then I ran into the serialising problem. So I think that I'm going to try writing my own normaliser next.

Your comment pointed me in the right direction and I looked at a few other modules and came up with this: render($item->entity->field_network_specific_site_code->valu‌​e), which prints the label of this field nicely (also for multi values).

At first, I tried the view mode approach, but it doesn't quite fit my requirements. I'm aware that hard-coding field names is problematic, but this is a custom module and the data model is fixed. Anyway, I just tried $item->entity->field_example->value, but it doesn't return anything. No error messages in the log either.