Leaving from GA has few advantages. Due to the trend of the coastline you are about 100 miles further west leaving from GA than from S FL or SC. From GA you could head
east until north of the Abacos and then turn south. If you try a more direct route
you will be sailing against the Gulf Stream
a lot of the trip.
Charleston is much further east and could give you a good reach all the way but the Gulf Stream
could still be mostly against you. You could cut across at an angle to get past the stream but that adds distance to the trip. If you try to cut inside the stream you end up sailing down the coast and give up the easting you would get from SC.
I would pick Charleston as the better option.
Regarding time, if you are following the standard summer in Newport
in the islands or similar pattern then the time would be after hurricane
season but before winter
storm season. This, in my opinion is a gamble demonstrated by the history
of even well found boats that are occasionally lost
heading south in Oct-Nov. The odds are very good but hurricanes are not unheard of in Oct or even Nov and I have seen winter storms in Sept.
If you are choosing strictly on the basis of the best/safest weather then very late spring before hurricane
season kicks off is my opinion (I assume you know what opinions are worth). However, to get a good forecast
for a reasonable five day window should be doable most times these days, even in hurricane season.