Ants are tough to battle. DE (diatomatious earth) would be my first recommendation but that isn't that effective in a wet environment and ants are fairly good at surviving such things anyway. That said food grade DE won't hurt your AP system if you sprinkle it over the beds and might even provide some nice trace elements while discouraging the ants. I expect it would take days of a dry coating of DE over the top of the bed to even irritate them much though and I'm not sure if it would cause them to move out.
If you can get them to leave with the constant deep flood, then you might be able to use some barrier method to keep them out in the future. However most of these methods either require a sticky goop around the edges and all paths into the beds that would stop ants from crossing or like I do in my worm bins, I place the feet in containers of mineral oil to create a moat but then you have to keep those containers free of debris so the ants can't find a bridge and you also have to keep all other objects from touching and creating a path.
I'v had two attacks by fire ants in my wick bed over the last month, I did the flood trick for about two days and that seems to make them move on to somwhere else. I don't know if anything will kill them. I do believe after man is long gone, fire ants will inherit the earth or whats left of it. I did see a sign on a fence that stated that the property was protected by fire ants, only in Texas!
Here is a fire ant trick I plan to try (I've read about it and been told by some people I know and trust that it works but haven't tested it yet myself.)
Molasses. A spoon full of molasses per gallon in hot water mix up and spray on the ant mound and around where you want to keep them away. I've been told it works.
I do know that spraying molasses around you system is quite safe and actually beneficial. I use molasses heavily in brewing my worm teas and I will often spray a molasses/water mix on my plants which helps feed the plants while also deterring pests or even killing some.
OK I got to try that! It would just be to simple. Its funny how many soulutions to our problems are often the simplest.