Its already been done in possibly the cheapest possible car.
Sell an all electric car called a G-Wiz or Reva. Being electric there is little power left over for heating and cooling so they came up with putting the climate control in the seat. The big advantage is that the heating and cooling takes little energy and as a side product is almost instant.
Not only can you have what you want but you could be doing the environment ( and so your children) a favour too.
On a more practical note you might want to consider a solar car fan.
these us small solar cells to drive a fan. This stops the car from acting like a solar panel or green house and getting dreadfully hot.
To make this car seat work.
Perhaps the cooling mechanism from one of those dorm/car coolers attached to a 12 volt solar cell would actively cool the seat. The Solar cell might fill the window but that's good as it acts as a sun shade too. Notice you would still need the solar car fan to keep the car at at least ambient temperature. Alternatively ( depending on temperature required ) a lower tech solution would be a bottle with water and a suitable wick ( cloth). The evaporation of water would help cool the seat (while the water lasts).