@Perfect10 a friend currently boards with Julie at New Vintage and her horse gets regular turnout (I have not asked how many hours). So, the amount may vary by which horses, but it does not seem to be a blanket no turnout rule.
@oceanmountain I looked at Megan’s Alpharetta farm and she said she does not do night turnout, she rotates her horses in the available turnouts during the day.
So, the OP would definitely need to ask about how much turnout the horse would get at both places (well, any places really!) and decide whether it met their needs.
Three Bears Farm definitely does night turnout, and does a mix of local GHJA shows and A/AA shows, including Tryon, Aiken and Ocala. So that would be another option, and it has an indoor.
Melinda Sise at MDS Farm and Ann Sherbourne at A&H Show Stables are two additional Alpharetta options.
And Michael Newman took over Atlanta Hunt Club, but I’m not sure of the turnout situation there. But it would fit the bill as a hunter barn that does A/AA shows and goes to Florida I would think.