To get it to work using convection, you need to have something which will liquify under your peltiers, and then drip down into the tank, and become vaporized again by your water. Something like freon, basically, otherwise you might just as well use a solid copper rod to the same effect. Unless you’re getting your peltiers down to 216K or lower and put it under a pressure of 518 kPa or higher. That is generally speaking a bad idea to do for various reasons.
And why bother using it from a cold pack? The amonium nitrate in there actually endothermally changes to another substance, hence creating the cold. It is not actually a good conductor by any means, and you would have to keep adding ALOT of it to the system continuously.