I dislike this extremely. The "bab" sound is very unattractive. I've never liked
Barbara either, but the nickname
Babs is infinitely worse. The "bab" sound makes me think of insane asylums and drooping breasts. Anyway,
Babette is no better than
Babs.
I think
Betsy as a nickname for
Barbara is counter-intuitive and quite a stretch.