I know it's not the time of year most think of seat coolers but just got into this thread and wanted to throw out a few ideas...
Flojet - ITT Flojet is a maker of pumps that are a little more industrial and larger than other suggestions I've seen and probably cost a bit more, but they're quiet and have published pressure/flow ratings (you can e-mail them for a pressure/flow curve once you select your configuration) and they sell in both AC and DC applications. I'm a bit familiar with their product as one of the machines I work on at work uses these.
Taking the function of that machine and I saw a natural extrapolation into the seat chiller. I don't know about silicon (doesn't that have very good thermal insulation properties?), but I am familiar with Tygon (tygothane) tubing. My only concerns there would be preventing the tube line from getting squished off closed from sitting/leaning on it. A harder nylon tubing might be good from a heat transfer standpoint, but it is *hard* and might be a pea to a few princesses.
Using this pump you could pull water in from a cold sink and pump out to a tee. One branch of the tee has a flow control valve and circulates through the seat and dumps back into the cold sink. The other branch of the tee is the bypass and dumps back into the cold sink. If the pressure/flow requirements of the seat circuit don't line up with those of the flow control you can put the FCV on the bypass, but then the adjustment is intuitively backwards (open the valve to reduce seat flow and vice-versa).
Additionally I was thinking the ice/cooler as a cold sink isn't really necessary if the vehicle has A/C already. It is possible to run the A/C in a manner to minimize the FE hit, so go ahead and run it on recirculate mode on the lowest fan setting. Wrap a copper line around the evaporator coil and use that as your cold sink. Just tie one end of the copper to the pump inlet and take both your seat return and bypass lines into a tee on the other end of the copper line. Your evaporator will get cold but the refrigerant low pressure switch will cycle the compressor off to prevent freeze-up. Getting the compressor into this cycle-off mode minimizes compressor run-time but still keeps the evaporator cold, allowing your seat heater to work. Insulating the lines between the evaporator and the pump and in the bypass circuit should help keep the fluid cold so you aren't pushing heat onto the evaporator unnecessarily.
How about a loop of tube with a sleeve over it you could pull over each shoulder and run down your neck? Cooling your jugular will cool you off fast. Just need a loop on each side with a stiffener/wire in it so that it hold its form where you put it (nothing to choke you of course).
Side benefit of this system is it could work as a seat heater when the A/C is off and the heater is on. The hot air in the blower housing would heat the water which you could pump into your cold seat. Doesn't help until the heat is blowing, but would still be faster than waiting for the cabin to warm.