tl;dr: you don't want to run at 15VDC. Really. 12V is plenty.
There are a number of factors that come into play. To get the first one out of the way - as has already been said, you can't get 15VDC from a computer supply by using the -12V rail, and while it would seem to be simple to just use the +12VDC and the 3.3VDC rails in the same fashion, the fact that they have a common ground makes that not work. [One of the supplies will be a dead short. Not pretty.] The other thing to consider is what the efficiency curves say. Most, if not all, TEC will cool better at a lower voltage than maximum. Since you didn't give us the part number I can't check it, but using the 12708 (which looks pretty close to your stated specs) - running at 12V/6A is almost as effective for a lot less power (72W vs 112W). Remember, your heat sink has to reject all of the power you put into the TEC, plus the heat transferred from the cold side. You may find that the 40W difference matters. A lot. More is not always better. Use google to find the data sheet for your exact TEC (the number is on the "hot side") - you'll find you can run a bit lower on the voltage, and still get almost the same cooling.
First of all, TMI of the useless sort, not enoughof the useful sort.
That converter is made to operate lights and DC motors. It probably has little to no filtering of the output, and may use only a half wave rectifier. It seems to be pretty old.
What that means is, is that it doesn't put out a nice, clean 12VDC, but rather a pulsating voltage that approximates DC.
The low points in the pulses are low enough to trigger the low voltage warning on your cooler.
Adding the (big) capacitors smooths the pulses and holds the average voltage high enough to stay above the low voltage alarm.
The peaks on the pulses may well be over 16VDC. You have no way to measure them, and I can't measure it from here.
So, no guarantees that it won't kill your (presumably) expensive capacitors.
Cheaper and more certain would be to use a modern switching power supply rated for 12VDC and 5A. Check the tags on the cooler, or look it up online to be sure about the voltage and current.
Or, just use a cooler made for 110VAC when you've got an outlet to connect to.
The recommended converter puts out 6A at 12V. The model number is in the cooler hand book. Buying one of those is probably cheaper than playing games with expensive capacitors, and safer besides. Have you considered what could happen if you short those big capacitors?
Best Answer
Without a datasheet, I'd assume that 15 volts is the maximum voltage that should be used, and that, at that voltage, the device will draw 68 watts. If you supply a lower voltage, the Peltier device willl draw less current and less than 68 watts.