Do you need in the barrels every day?
1) Could you tarp them in such a way that undoing the tarp and dumping the water is a near single motion? “Unclip, raise, lower, reclip”
2) Could you fill the recessed top with something like a floor leveler? Or just clay.
3) Could you get a large sponge or sponge material cut to size and glue it to the top, the sponge wicks up the water and increases surface area for evaporation. No standing water for breeding and just takes a quick depression with a cut board circle to squeeze out excess water.
4)Is the top lid thick enough that you could cut a piece out of the rim so the water passively runs out without it also getting into what is in the barrel.
That’s what I got so far, good luck.