To make a two stage Pelter cooling system work you will need to have the higher wattage one cooling the lower wattage one. Overall cooling efficiency will be lower but the cold side temperature limit will be lower in exchange.
Depending on the cooling efficiency of the higher wattage unit you may need a 2:1 or possibly a 3:1 wattage ratio between them which means that your present 150 watt cooler would be the large one and the smaller second one would need to be in the 50 - 75 watt rage if they are both of reasonable efficiency. If not you smaller one may have to be down in the 40 - 50 watt range.
Also the larger cooler will need to have a good sized heat sink being that it will be trying to dump as much as 300 watts of heat energy at any one time when the system is running.
As far as making ice it should be easy if your box is insulated well enough and you can keep your outside heat sink temperature as close to ambient temperature as possible.
One issue I see is that using a common computer power supply is that its wattage rating is intended to be spread out over all of its outputs and not just one output such as the 12 VDC circuit. Trying to pull all or nearly all of the rated wattage off of the 12 VDC output alone will cause it to have a short life expectancy and typically very poor voltage regulation when it is working.
Ideally you should have dedicated power supply if you are planning to be running higher wattages on a single output system.