A day trip really doesn't give you a sense of the place, so I'd not bother doing this unless you're going to at least stay overnight.
However, if you decide to do so, you should know that one parent travelling with a child without the other parent present, should have custody papers showing sole custody or a notarised letter from the other parent giving permission to take your child out of the country. Sometimes they ask for it, most of the time not, but it's worth having in case.
Makes sense! There's so much to do and see in the Bahamas -- with over 700 islands -- that a day trip really doesn't do it justice. Definitely plan another trip and come and see the real Bahamas.