Basically the whole thing felt inappropriate to me. They should be able to get it together enough to just hand over their child. I was concerned about getting involved in that way. The drop off's and pick ups now are not an issue as he spends his day with me so it is not awkward for me/the child as just the other parent picks up.
It was much easier to say "no" as he was home sick for the day but even if he wasn't if would have been weird. I think they can do more for their kid if they put on their brave faces and do the exchange. If the parents can be human/normal in front of each other then maybe this child won't feel so many conflicting loyalties.