KILLorBE,
I'm a 40 year old electrical engineer. I know very well when a cap does or does not matter.
The fact that for all practical purposes a TEC draws a DC current is precisely the reason the cap will have no value. An ideal capacitor has absolutely no significance in a truly DC circuit. It is effectively 'not there'. (Any non-idealities of an actual capacitor are irrelevant at best.)
The fact that the voltage to the TEC will come up a bit more slowly at power up with the cap in place is of no value, and is ultimately irrelevant to the cooling of the TEC.
The reason a cap gives an RC car more punch, is because that is not a DC situation. (The voltage and current of the motor are changing.)