I understand you wanting what is best for your children and to take the avenue that will yield as little drama as possible.
I guess the one thing I would say is think about how unglued he becomes when you do one thing different (ex: not greeting him) and how dramatic he becomes (stomping off). He is about to get the shock of his life... being served divorce papers and realizing all at the same time you are 110% aware of exactly what has been going on, for how long, where and with who. You really can't predict how he will react and despite your desire for a drama free exchange it might not pan out that way based on his behavior when he doesn't get his way.
At least at the apartment (and I agree that would be beyond delicious) he will have a bit of time and space to react and your children will not have to witness any outlandish behavior.
Also, if he is served at the apartment his little "friend" might also get the shock of her life that she is indeed invested with a very married man and father. Talk about getting it at both ends!
