I would try sweet peppers that are not bell pepper types. I live in a different region, Wisconsin, but I had very good luck with Carmen.
I bet a sweet banana pepper would do well. There are lots of hybrid types you can buy as plants and seeds that finish pretty early, like Pageant and Lola. I am trying both this year.
Another type of sweet pepper that I have read about and watched people talk about on Youtube that does well in most regions is the Nardello/Jimmy Nardello's. This is an Italian frying sort.
Another type of frying sort is the shishito pepper, a Japanese frying sweet pepper, and there are some hybrids for that, like Mellow Star and Takara. I am trying both the Nardello and Takara this year, pretty excited for them.
Another intriguing option for you is heatless tropical hot peppers.
Here are some suggestions for this
Roulette, Habanada and Zavory (Heatless habanero style)
Sweet Datil (heatless sweet Datil style)
Sweet Cayenne
Tobago Seasoning, Grenada Seasoning, have several hundred SHU but are within the 1000 SHU cut off and might do well in the Alabama climate.
I think these last ones would be like the Aji Dulce being talked about.