This has been an issue for a long time. Either military, racing, or film costumes, a great deal of research has gone into developing a way to cool the body in these types of situations.
The problem is the human body can generate a lot of heat, and finding a way to absorb or dissipate this heat isn't easy.
The pictured suit above is your best alternative, given all that is currently out there. I am not really a fan of the ice pack vests, I much prefer the circulating fluid systems. I have worn both in creature suits for film or TV and the circulating system is much more consistent and controllable. You really don't need, or necessarily want 32-40 degree materials against the skin, you really only need 50 to 60 degrees.
Wet ice is the best for this, as it gives up more energy than dry ice and peltier devices (cooling chips) are far too inefficient to have any affect unless you have a huge number of them, and then the power source and unit are way too big to be practical.
This problem of "cooling power" is such that for film we didn't bother with a self contained unit. Because by the time you got it small enough to fit on a costume (even with a back pack) it would either get too heavy or only last a hour at best.
I developed a "cool suit" for my friend Kevin Peter Hall for the first Predator, since at that time there was almost nothing commercially available. The few that were, where very expensive or not designed with costume work in mind.
Kevin was very worried about over heating in the suit in the jungles of Mexico, and Stan Winston's guys had no idea what Kevin was talking about when I suggested he ask them for one. Eventually I started making them for rent, and over the years they have been used on hundreds of projects. Which led to this:
(Harvey Hubert worked for me and built many of our first pump systems)
Now there are many commercially available "coolsuits" and many production companies just buy them for the run of the show.
While our vest was a very different design, using "heat sealed" channels in an open "radiator design", verses clear tubing in a serpentine design, the tube suits work pretty well.
But most commercial units have a big drink cooler for the ice reservoir and that isn't practical for a walk around suit.
If you can get a screw top plastic bottle and drill holes in the lid to run your pump-to-vest lines in and out of the bottle. (one hose in and a second out) You fill it half way with water and freeze it. Then when you want to use it, fill the rest of the space with water and hook up you hoses from the pump to the vest to the bottle. (make sure the hoses are long enough to be completely in the water but not frozen into the ice.) This way you have a small compact "cold battery". You can glue the hoses into the lid and just screw on/ off containers, or you can use quick disconnects(I like CPC fittings) mounted into the lid and the hoses.
To assist the water flow and keep it from growing, add a couple of drops of iodine and wetting agent to your water. This will keep it from "growing" and the wetting agent makes the water wetter, so it flows with less resistance.
Don'y drink the water from you pump/vest system, as it either has iodine in it, or if not, the water will grow algae (or other things) in the vest, and it can make you sick.