I've done all three, several times, and the answer to your question depends on what you want.

This is an overnight flight so I imagine you'll want to sleep as much as possible. VS has an excellent bed, with a flip-over mattress. The new BA Club seats are also fantastic, and I say this having very recently come off an overnight flight from JFK. AA F is good too, though the cushion is a bit thin.

Catering is really hit and miss. I've had good and bad on all three. AA is probably the underdog.

If you are up for a James Bond-type experience of drinking at the bar and chatting up quasi-hot flight attendants, you'll be happy on VS, BA might be ok, and AA will be like picking up your grandmother's friends.