If it's dry out, it will be faster than if it was humid.  

In an area of high moisture and humidity like that, the foam will hold the water.

I would not do this in a hot humid climate where it would be expected the windows would be open.

You are in the humid category so the vapor barrier should be on the outside (exterior) of the of the insulation. 

Also, any recommendations on what material to use down there - it is right on the beach and gets super humid.  

Its a rental,  so it should be a full one piece fiberglass install,  and Add a ventilation fan and ideally hook it up to an automatic humidity sensor switch  (good upgrade too for any bathroom that doesn't have it).    

Deborah BurianThe building science site has many useful case studies and even though many of them were conducted in the more humid environment of New England, the temperature swing from winter to summer will be comparable.When you say the house does not have a vapour barrier at the exterior wall, do you mean house wrap (tar paper in those days) or an actual vapour barrier (which should be on the warm side of the wall)?

3) paint and carpetAlso if it’s a humidity deal try a dehumidifier If that doesn’t work ... hire a priest and perform exorcism 😉

We don't have harsh winters, wet humid conditions, or basements to contend with.

I would told that the drywall is throughout the United States, however it is only toxic and reactive in high humidity.