Chemist here, just musing @ breakfast, so I am not backing any of that stuff with literature research.
I understand you would like to have a permanent cooling effect. Unfortunatly, materials that undergo conformational/phase transition processes do not require a permanent feed of thermal energy e.g.
A + dT --> A' conversion with heating
A' -> A + dT fall back to original state with cooldown
with dT temperature difference (thermal energy)
and A and A' the respective states of the material.
As such you would need a material that uses the thermal energy for some other effect e.g. radiation of light, permanent conformational movement creation of electric potential etc. The trouble I see here is that the effiency of energy conversion is fairly low with the current polymers. And the price tag.
Ok, here is a crazy idea, but not sure if it is not impractical or even stupid (; . Maybe use of superadsorbing polymers with a specific thermoresponsive behaviour somewhere around the working temperature of the athlete would give you a higher cooling effect. Once the athlete heats up, the polymer becomes more hydrophilic, allowing for more sweat to enter the fabric. Hopefully, the wet fabric would have a better thermal energy flow than the dry fabric. Basically, you make a supersoaking sweatshirt. But then again, maybe you just use a hydrophilic fabric?