I like
May a lot and
Mae not at all.
May never seems like a month name to me, for some reason - I'm more likely to think of the flowering shrub.
I'd certainly avoid
June as a name for a
June baby with an
August brother! One way around the issue would be to name the baby something that
May can comfortably be a nn for:
Mary,
Maia and
Margaret, as you say, and also
Mabel.