Deep water culture uses a polystyrene "raft" which has holes which hold the baskets containing the hydroton and rock wool for the plants. This floats on top of the nutrient solution. The roots shouldn't rot if the water is well aerated so normally there is an air bubbler below.
So, I wonder if it's feasible to have a small raft in your tank with a few basil plants. You don't want the raft to cover the entire surface as that's your oxygen exchange occurs. Normally they say that the container has to be black to prevent algae growth but if you also had some plants in the tank, it might be enough to suppress the algae. The plants in the basket should out compete the immersed plants though.
The other issue is whether the fish will eat the dangling roots ...