Ahh, good old chemistry. (I'm a chemist)
Yeah, you can force larger amounts of gas into a system by increasing pressure via Henry's law or swapping out what you are pumping into the system (oxygen versus air).
The surface area doesn't apply in a bait tank, as you are increasing the surface area by using an air stone and a bubbler. That is increasing your surface area exponentially, and depends on the size of the bubbles (smaller is more area of contact).
If you use a sealed bait tank that will hold more than 1 atm of pressure you can increase it as well.
But, as soon as the tank is opened, all the gas dissolved will bubble out and form a new equilibrium, resulting in dead fish if you try and increase your carrying capacity via this route. Also the dissolved gasses inside your bait will come out of solution much like the bends on a diver.
Easiest and most practical way to keep more bait alive is maintain the water temp lower, as the carrying capacity for O2 in the water is greater at a lower temp. Too cold and fish get shocked when hitting warm fishing water though.
If you cannot do this, your only route is to reduce the number of baitfish within the tank. Aerators will only help up to the carrying capacity of the water.
One thing you could try, that would help and be less hassle than ice is to use a white or mylar coated bucket (mylar might annoy other fishermen though), and if you have a 12v bubbler is to hook up a computer cooling fan on the top of the tank and make a small hole for the air to escape. The air blows into the tank and evaporates water, cooling the water in the tank. Called a "swamp cooler" and may or may not help much, depending on the humidity. Higher humidity, less help.
Computer fans run on 12v DC and can have an affect on keeping the water temp down, just have to be careful about splashing.
My suggestion is to carry less bait. Fun to think about other ways though.