Are you running the AP with a hood, or are you topless? If you're using the stock hood, you really need to ventilate the display tank. The hood fan is only really useful for heat exhaust, not ventilation. One possibility is to keep the feeding door open, or, mount a fan on the side of the hood though a cutout blowing in. That may help with your gas exchange, and can even help cut your temps back if you're running high. You will increase your evaporation, but it may help.
Another quick fix would be to lower the water level in the rear compartments so the water coming over the overflow will splash down. It's simple, and the "waterfall" effect will help in degassing as well. If you're topless, adding a sump will increase your water volume, but probably won't improve gas exchange as you will still have a roughly the same ratio of volume to surface area. Setting it up as a refugium, though, can help to a degree.
A skimmer will help, even if it's a small skimmer like the sapphire skimmer in a rear chamber with the top down. Like Matthew, though, I've found opening the window a bit worked wonders, and I now run a pH of 7.8-8.0 or 8.1 with an alkalinity of 2.8.
Kalk would be a good solution, but unless you're topless, there's just not enough evaporation.