Hey MadHacker,
I believe your idea could work if you made a few simple changes (as others have suggested). Like this...
Split the TEC into a cold-loop and hot-loop configuration. Get rid of the raddy on the cold-loop side (put it in parallel with the raddy on the hotside if you want to use it).
Coming up with a control circuit is your next challenge! That's a lot of power to control and regulate (and TECs don't like PWM).
Good luck...

(keep those posts and ideas coming)